Modeling calcium dynamics in neurons with endoplasmic reticulum: existence, uniqueness and an implicit-explicit finite element scheme

Abstract

Like many other biological processes, calcium dynamics in neurons containing an endoplasmic reticulum is governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using the calcium model as an example of this class of ODE-flux boundary interface problems, we prove the existence, uniqueness and boundedness of the solution by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard’s existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in H¹ norm is obtained. Numerical experiments illustrate the theoretical results.

1. Introduction

In a variety of applications, particularly in biology, spatio-temporal dynamics can be described by diffusion-reaction systems. Intuitively, and from an energy consumption perspective, such systems appear optimal but usually have the drawback of producing slow, short-range, and potentially inefficient communication pathways. In order to overcome these drawbacks, active and energy-consuming processes are introduced by biology on two-dimensional manifolds, i.e. interfaces which separate multiple domains. Such processes, in the biological context, are channels, pumps, and receptors capable of exchanging specific ions across the interfaces and are mathematically described by systems of ordinary differential equations (ODEs), nonlinearly coupled to the domain equations, see [1, 2, 3, 4, 5, 6, 7, 8]. Given the ubiquitous nature of this modeling approach, we study a mathematical model consisting of diffusion-reaction equations in Ω_c and Ω_e (see Fig. 1 for a 2D case), coupled by nonlinear dynamic boundary conditions involving an ODE system on the interface Υ. To provide context to this ODE-coupled system of partial differential equations (PDEs), we chose cellular calcium dynamics as a leading example. Calcium dynamics are among the most important regulators in neurons and their three-dimensional (3D) spatio-temporal dynamics have been shown to play a critical role in cellular function, learning, and many neurodegenerative diseases, see [9, 3, 10, 11, 12]. Calcium dynamics is further controlled by the 3D organization of cells, see [13, 10, 11, 14], thus making calcium in neurons a prime candidate for the problems studied in this paper.

Figure 1:

Domain Ω = Ω_c ∪ Υ ∪ Ω_e.

The geometry structures of neurons can be found in [10, 14]. Neuron has a tubular organelle - endoplasmic reticulum (i.e. ER), which can be thought of as a cell-within-a-cell. For simplicity, without loss of generality, we assume the domain of the whole cell in 3D, including ER, is simply connected without holes, with smooth boundary; so does the domain of ER. Also there are no other organelles in this cell, boundaries (i.e. membranes) of ER and the cell have no contact. A cross section of an axon of the simplified neuron cell can be shown by Figure 1, where Ω_e is the region of ER lumen; Υ is the ER membrane; Ω_c is the region of cytosol; ∂Ω is the plasma membrane.

Here, let Ω be a bounded domain with C^1,β boundary in ${ℝ}^n$, n = 2, 3, Υ be the C^1,β interface, 0 < β < 1. We define

• $u:{\overline{\mathrm{\Omega }}}_c\times [0,T]\to ℝ$ as the Ca²⁺ concentration in cytosol,

• $b:{\overline{\mathrm{\Omega }}}_c\times [0,T]\to ℝ$ as the concentration of a buffer interacting with u,

• $u_e:{\overline{\mathrm{\Omega }}}_e\times [0,T]\to ℝ$ as the Ca²⁺ concentration in ER,

where [0, T] is the time domain. We define the PDE model by (1) to (9)

$${\partial }_{t}u-\nabla \cdot \left(D_c\nabla u\right)=f(b,u)\text{on}{\mathrm{\Omega }}_c\times (0,T]$$ (1)

$${\partial }_{t}b-\nabla \cdot \left(D_b\nabla b\right)=f(b,u)\text{on}{\mathrm{\Omega }}_c\times (0,T]$$ (2)

$${\partial }_t{u}_e-\nabla \cdot \left(D_e\nabla u_e\right)=0\text{on}{\mathrm{\Omega }}_e\times (0,T]$$ (3)

∂_t is the partial derivative in time, $f(b,u)=K_b^{-}\left(b^0-b\right)-K_b^{+}bu$ is the reaction term which models storing and releasing Ca²⁺ in cytosol, see [2] for more details, and $D_c,D_b,D_e,K_b^{\pm },b^0$ are positive constants. Denoting the outer normal derivative by ∂_n, direction of the unit vector n depends on which domain the function lies, the boundary conditions describing fluxes across the interfaces are

$$D_c{\partial }_{n}u=J_{l,p}-J_N-J_P\text{on}\partial \mathrm{\Omega }\times (0,T]$$ (4)

$$D_c{\partial }_{n}u=J_R+J_{l,e}-J_S\text{on}\mathrm{\Upsilon }\times (0,T]$$ (5)

$$D_e{\partial }_n{u}_e=J_S-J_R-J_{l,e}\text{on}\mathrm{\Upsilon }\times (0,T]$$ (6)

$$D_b{\partial }_{n}b=0\text{on}\partial \mathrm{\Omega }\cup \mathrm{\Upsilon }\times (0,T]$$ (7)

with initial data

$$u(0,x)=u_0(x),b(0,x)=b_0(x)\text{on}{\mathrm{\Omega }}_c;u_e(0,x)=u_{e,0}(x)\text{on}{\mathrm{\Omega }}_e$$ (8)

$$c_1(0)=c_1\left(0,u_0(x)\right),o(0)=o\left(0,u_0(x)\right),c_2(0)=c_2\left(0,u_0(x)\right)\text{on}\mathrm{\Upsilon }.$$ (9)

The choice of the fluxes J_N, J_P, J_l,p, J_R, J_S, J_l,e in (4)–(6) is motivated by the calcium models previously studied in [15, 16, 2, 17, 18, 7]. They describe the pumps that exchange Ca²⁺ across ∂Ω and Υ. Ca²⁺ flux crossing plasma membrane (∂Ω) is governed by PMCA pumps (J_P), NCX pumps (J_N) and leak channels (J_l,p); Ca²⁺ flux crossing ER membrane (Υ) is controlled by RyR channels (J_R), SERCA pumps (J_S) and leak channels (J_l,e), see [9, 10] for introduction, and [2] for an illustration. Below we list the fluxes under consideration. On Υ, the flux through RyR channels (or Ryanodine receptors) is

$$J_R=C_1^{e}P(t,u)\left(u_e-u\right),$$ (10)

where $C_1^e$ is a positive constant and for u ≥ 0, P(t, u) ∈ [0, 1] is the probability that RyR channel is open:

$$P\text{(}t\text{,}u\text{)=1-}{c}_{\text{1}}\text{(}t\text{)-}{c}_{\text{2}}\text{(}t\text{),}$$ (11)

and c₁, c₂ come from the ODE system

$$\left[\begin{array}{c}{c}_1^{\prime }\\ o^{\prime }\\ c_2^{\prime }\end{array}\right]=\left[\begin{array}{ccc}-u^4{k}_a^{+}-k_a^{-}& -k_a^{-}& -k_a^{-}\\ -u^3{k}_b^{+}& -u^3{k}_b^{+}-k_b^{-}& -u^3{k}_b^{+}\\ -k_c^{+}& -k_c^{+}& -k_c^{+}-k_c^{-}\end{array}\right]\left[\begin{array}{c}{c}_1\\ o\\ c_2\end{array}\right]+\left[\begin{array}{c}{k}_a^{-}\\ u^3{k}_b^{+}\\ k_c^{+}\end{array}\right],$$ (12)

where $k_a^{\pm }$, $k_b^{\pm }$, $k_c^{\pm }$ are positive constants, and obviously {c₁, o, c₂} depend on position due to u, but for simplicity, the spatial variable is hidden, the initial values of {c₁, o, c₂} are non-negative and c₁(0) + o(0) + c₂(0) ≤ 1, see [2] and [17] for details. Next, we define the flux through SERCA pumps

$$J_S=C_2^e\frac{u}{\left(K_s+u\right){\varphi }_m\left(u_e\right)},J_{l,e}=C_3^e\left(u_e-u\right),$$ (13)

where K_s, $C_2^e$, $C_3^e$ are positive numbers, ϕ_m(·) is defined as

$${\varphi }_m(x)=\{\begin{array}{ll}m/2,\hfill & \text{if}x\le 0\hfill \\ m^6/\left(2m^5-5m^2{x}^3+6m{x}^4-2x^5\right),\hfill & \text{if}0<x<m\hfill \\ x,\hfill & \text{if}x\ge m\hfill \end{array}$$

where m > 0 can be any small number. On ∂Ω we define

$$J_P=C_1^c\frac{u^2}{K_p^2+u^2},J_N=C_2^c\frac{u}{K_n+u},J_{l,p}=C_3^c\left(c_o-u\right)$$ (14)

where $C_1^c$, $C_2^c$, $C_3^c,K_p,K_n$ are positive constants, c_o is the extracellular Ca²⁺ concentration, we assume it is a positive constant, however c_o can also be a positive bounded function. J_N and J_P are commonly used first and second order Hill equations, see [2].

An analysis of linear parabolic equations in two adjoining domains, coupled with nonlinear but nondynamic boundary conditions, can be found in Calabrò [19]. An implicit DG method in [20, 21] was developed to solve such problems that originated from mass transfer through semipermeable membranes. Poisson equations coupled with nonlinear dynamic boundary conditions in [1, 6, 8], arising in electrodynamics on membranes are analyzed. A numerical method in [22] based on a boundary integral formulation, together with an implicit-explicit scheme, is proposed to solve these electrical activity problems, also implicit-explicit partitioned time stepping for a parabolic two domain problem is considered in Connors[23]. Both motivate us to design our scheme. However the analysis and numerical methods in [1, 19, 20, 21, 22, 6, 8] cannot be applied to our model directly, since the reaction term f(b, u), and the interface conditions, especially the non monotonic probability function P(t, u) involving ODE systems, are not globally Lipschitz continuous. Previous work on Ca²⁺ models focused on simulations employing numerical methods such as the Finite Volume Method in [16, 2], the Finite Element Method in [24, 18], but most of them are fully implicit and less efficient. Despite the importance of calcium models, to our knowledge, a thorough study for the wellposedness, fast numerical methods and convergence analysis have not been done. The goal of this paper is to prove existence and uniqueness, to retrieve bounds for solution of the model (1)–(9), then to obtain error estimates for an implicit-explicit FEM scheme which is more efficient and easy to parallelize, since equations (1)–(3) are fully decoupled for each time step. Unlike the methods used in [1, 19, 6, 8], our proof of wellposedness is inspired by Picard’s existence theorem, based on the fundamental solution of parabolic operator. The FEM scheme is an Euler method implicit for the parabolic operator and explicit for the nonlinear parts. A key part of the error analysis is to construct the elliptic projection, similar to the work in Cangiani[20] and Douglas[25], but incorporating the dynamic ODE system. So that the wellposedness of the associated nonlinear problem needs to be addressed.

The paper is arranged as follows: Necessary lemmas are provided in Section 2. The existence, uniqueness and boundedness of the solution for the model equations (1)–(9) are proved in Section 3. Section 4 is devoted to the H¹ error estimate for the Galerkin projection which is based on a nonlinear problem in variational form. In Section 5, the error analysis of the semi-discrete Galerkin method is carried out. The convergence rate in H¹ norm for a fully discrete implicit-explicit FEM scheme is obtained in Section 6. Numerical tests in Section 7 validate the theoretical results. Conclusions are drawn in Section 8.

2. Lemmas

In this section, we introduce some notations and lemmas. We define Q_c = (0, T] × Ω_c, Q_e = (0, T] × Ω_e. In the following M is a positive constant and function u being bounded by M means |u| ≤ M.

We then show that the open probability function P(t, u) (see eq. (11)) is continuous with respect to u if u ∈ C(Υ × [0, T]) and 0 ≤ u ≤ M.

Lemma 2.1.

Let u₁, u₂ ∈ C(Υ × [0, T]) be non-negative functions bounded by M and P(t, u) be defined as (11)–(12). We then have

$$\left|P\left(t,u_1\right)-P\left(t,u_2\right)\right|\le K\left(\left|u_1(0)-u_2(0)\right|+{\int }_0^t\left|u_1(s)-u_2(s)\right|\text{d}s\right),$$ (15)

where K is a positive constant that depends on M, but doesn’t depend on u₁, u₂.

Proof.

Let $\overrightarrow{q}(t)={\left[c_1,o,c_2\right]}^T$, $\overrightarrow{f}(u)={\left[k_a^{+},u^3{k}_b^{-},k_c^{-}\right]}^T$, and A(u) be the coefficient matrix in (12). The ODE system can then be written as

$$\frac{\text{d}\overrightarrow{q}}{\text{d}t}=\mathbf{A}(u)\overrightarrow{q}(t)+\overrightarrow{f}(u).$$ (16)

For given u₁, u₂, the corresponding integral equations are

$${\overrightarrow{q}}_i(t)={\int }_0^t\mathbf{A}\left(u_i\right){\overrightarrow{q}}_i(s)\text{d}s+{\int }_0^t\overrightarrow{f}\left(u_i\right)\text{d}s+{\overrightarrow{q}}_i\left(0,u_i(0)\right),i=1,2.$$

Let $\overrightarrow{e}(t)={\overrightarrow{q}}_1(t)-{\overrightarrow{q}}_2(t)$, ${\overrightarrow{f}}_e={\overrightarrow{f}}_1\left(u_1\right)-{\overrightarrow{f}}_2\left(u_2\right),{\mathbf{A}}_{\mathbf{e}}=\mathbf{A}\left(u_1\right)-\mathbf{A}\left(u_2\right)$ and ${\overrightarrow{q}}_{0,e}={\overrightarrow{q}}_1\left(0,u_1(0)\right)-{\overrightarrow{q}}_2\left(0,u_2(0)\right)$, we have

$$\overrightarrow{e}(t)={\int }_0^t\mathbf{A}\left(u_1\right)\overrightarrow{e}(s)\text{d}s+{\int }_0^t{\overrightarrow{f}}_e\text{d}s+{\int }_0^t{\mathbf{A}}_{\text{e}}{\overrightarrow{q}}_2(s)\text{d}s+{\overrightarrow{q}}_{0,e}$$

$$\Vert \overrightarrow{e}(t){\Vert }_1\le {\int }_0^t{\Vert \mathbf{A}\left(u_1\right)\Vert }_1\Vert \overrightarrow{e}(s){\Vert }_1\text{d}s+E_h(t)$$

where || · ||₁ is the 1-norm and

$$E_h(t)={\int }_0^t{\Vert {\overrightarrow{f}}_e\Vert }_1\text{d}s+{\int }_0^t{\Vert {\mathbf{A}}_{\mathbf{e}}\Vert }_1{\Vert {\overrightarrow{q}}_2(s)\Vert }_1\text{d}s+{\Vert {\overrightarrow{q}}_{0,e}\Vert }_1.$$

With E_h(t) being non-negative and non-decreasing, by Gronwall’s inequality, we have

$$\Vert \overrightarrow{e}(t){\Vert }_1\le E_h(t){\int }_0^t{\Vert \mathbf{A}\left(u_1\right)\Vert }_1\text{d}s$$ (17)

where u₁, u₂ are bounded and the initial condition of the ODE system is Lipschitz continuous, so that ${\overrightarrow{q}}_1$, ${\overrightarrow{q}}_2$ are bounded and

$$\Vert \overrightarrow{e}(t){\Vert }_1\le K\left(\left|u_1(0)-u_2(0)\right|+{\int }_0^t\left|u_1(s)-u_2(s)\right|\text{d}s\right).$$ (18)

With $\left|P\left(t,u_1\right)-P\left(t,u_2\right)\right|\le \Vert \overrightarrow{e}(t){\Vert }_1$, the proof is completed. □

For brevity, from equation (4) and (6) we define

$$g_c(u):=D_c{\partial }_{n}u\text{on}\partial \mathrm{\Omega },g_e\left(u,u_e\right):=D_e{\partial }_n{u}_e\text{on}\mathrm{\Upsilon }.$$

With Lemma 2.1 we get

Lemma 2.2.

Let u₁, u₂ ∈ C(∂Ω_c × [0, T]) and u_e1, u_e2 ∈ C(Υ × [0, T]) be non-negative functions bounded by M, then we have

$$\left|g_c\left(u_1\right)-g_c\left(u_2\right)\right|\le K_1\left|u_1-u_2\right|,$$

$$\left|g_e\left(u_1,u_{e1}\right)-g_e\left(u_2,u_{e2}\right)\right|\le K_2\left(\left|u_1-u_2\right|+\left|u_{e1}-u_{e2}\right|+\left|u_1(0)-u_2(0)\right|\right)$$

$$+K_2{\int }_0^t\left|u_1(s)-u_2(s)\right|\text{d}s,$$

where K₁, K₂ are positive constants that depend on M, but do not depend on u₁, u₂ and u_e1, u_e2.

Lemma 2.3.

[25] Let D be the domain with appropriate boundary, there exists a positive constant C_T such that for 0 < ϵ ≤ 1

$$\Vert v{\Vert }_{L^2(\partial D)}\le C_T\left(ϵ\Vert \nabla v{\Vert }_{L^2(D)}+{ϵ}^{-1}\Vert v{\Vert }_{L^2(D)}\right),v\in H^1(D).$$ (19)

Remark 1.

Lemma 2.2 together with smooth enough g_c, g_e and (20) to (22)

$$g_c(0)\ge 0,g_c\left(c_o\right)\le 0,g_c(u)\le C_c,\mathit{\text{for}}u\ge 0,c_o,C_c>0$$ (20)

$$g_e\left(0,u_e\right)\le 0,g_e(u,0)\ge 0,\mathit{\text{for}}u,u_e\ge 0$$ (21)

$$-K_5{u}_e-K_6\le g_e\left(u,u_e\right)\le K_{3}u+K_4,\mathit{\text{for}}u,u_e,K_i>0,i\ge 3$$ (22)

can also be viewed as conditions for the analysis in following sections. So that the analysis and numerical method can be applied to models with more general interface/membrane fluxes.

3. Existence, Uniqueness and Boundedness

3.1. Fundamental Solution

For completeness, we recall the definition and properties of the fundamental solution of the parabolic operator, for more information see [26, 27]. We define the operator L as $Lu={\partial }_{t}u-\mathcal{D}\mathrm{\Delta }u$, where $\mathcal{D}>0$ is some coefficient. Then the fundamental solution of L is

$$\mathrm{\Gamma }(t,x;\tau ,\xi )={[4\pi \mathcal{D}(t-\tau )]}^{-n/2}{e}^{-\frac{|x-\xi {|}^2}{4\mathcal{D}|t-\tau |}}.$$

For any x, ξ in ${ℝ}^n$, n = 2, 3 and 0 ≤ τ < t ≤ T, the fundamental solution has bounds

$$|\mathrm{\Gamma }(t,x;\tau ,\xi )|\le \frac{K_0}{{(t-\tau )}^{\mu }}\frac{1}{|x-\xi {|}^{n-2+\mu }},0<\mu <1$$ (23)

$$\left|\frac{\partial \mathrm{\Gamma }(t,x;\tau ,\xi )}{\partial v(t,x)}\right|\le \frac{K_0}{{(t-\tau )}^{\mu }}\frac{1}{|x-\xi {|}^{n+1-2\mu -\gamma }},1-\gamma /2<\mu <1$$ (24)

where K₀ is a constant independent of (t, x) and (τ, ξ). Let D be an open bounded domain with C^1,β boundary. The second initial boundary value problem is given by

$$\{\begin{array}{ll}Lu(t,x)=f(t,x)\hfill & (t,x)\in D\times (0,T]\hfill \\ {\partial }_{n}u=g(t,x)\hfill & (t,x)\in \partial D\times (0,T]\hfill \\ u(0,x)=u_0(x)\hfill & x\in \overline{D}\hfill \end{array}$$ (25)

where f, g, u₀ are any given functions. The solution to (25) is:

$$\begin{gathered} u(t,x)={\int }_0^t{\int }_{\partial D}\mathrm{\Gamma }(t,x;\tau ,\xi )\psi (\tau ,\xi )d\xi d\tau +{\int }_D\mathrm{\Gamma }(t,x;0,\xi )u_0(\xi )d\xi \\ +{\int }_0^t{\int }_D\mathrm{\Gamma }(t,x;\tau ,\xi )f(\tau ,\xi )d\xi d\tau , \end{gathered}$$ (26)

where ψ can be obtained from solving the following integral equation:

$$\psi (t,x)=2{\int }_0^t{\int }_{\partial D}\frac{\partial \mathrm{\Gamma }(t,x;\tau ,\xi )}{\partial v(t,x)}\psi (\tau ,\xi )d\xi d\tau +2H(t,x),$$

$$H(t,x)={\int }_D\frac{\partial \mathrm{\Gamma }(t,x;0,\xi )}{\partial v(t,x)}{u}_0(\xi )d\xi +{\int }_0^t{\int }_D\frac{\partial \mathrm{\Gamma }(t,x;\tau ,\xi )}{\partial v(t,x)}f(\tau ,\xi )d\xi d\tau +g(t,x).$$

Then, to get an explicit expression of ψ, same as in [27], we define

$$Q:=\frac{\partial \mathrm{\Gamma }(t,x;\tau ,\xi )}{\partial v(t,x)},Q_{j+1}:={\int }_0^t{\int }_{\partial D}Q(t,x;s,y)Q_j(s,y;\tau ,\xi )dyds,$$

where j ≥ 1 and Q₁ = Q. The solution ψ(t, x) has the explicit form

$$\psi (t,x)=2H(t,x)+2{\int }_0^t{\int }_{\partial D}R(t,x;\tau ,\xi )H(\tau ,\xi )d\xi d\tau .$$ (27)

R(t, x; τ, ξ) is denoted by

$$R(t,x;\tau ,\xi )=\sum _{j=1}^{\infty }{Q}_j(t,x;\tau ,\xi )$$ (28)

and R(t, x; τ, ξ) has the following bound

$$|R(t,x;\tau ,\xi )|\le \frac{K_1}{{(t-\tau )}^{\mu }}\frac{1}{|x-\xi {|}^{n+1-2\mu -\gamma }},1-\gamma /2<\mu <1,$$ (29)

while K₁ does not depend on the variables, see [26] for a proof.

3.2. Wellposedness

For brevity we define the parabolic operators as:

$$L_{c}u={\partial }_{t}u-D_c\mathrm{\Delta }u,L_{b}b={\partial }_{t}b-D_b\mathrm{\Delta }b,L_e{u}_e={\partial }_t{u}_e-D_e\mathrm{\Delta }{u}_e,$$

the corresponding fundamental solutions are Γ_c, Γ_b, Γ_e, and R in (27) are represented by R_c, R_b, R_e. We denote C_1,2(Q_c) as the space of functions with two continuous spatial derivatives and one continuous time derivative on Q_c. The boundary conditions for u are defined as $B_{c}u:={\partial }_{n}u$ on ∂Ω, $B_{e}u:={\partial }_{n}u$ on Υ. In this section, we derive the bounds for solution (if it exits) of (1)–(9) in Theorem 3.1, then show the existence and uniqueness of the solution in Theorem 3.2.

Theorem 3.1.

Assume {u, b, u_e} is a solution of (1)–(9), and $u,b\in C\left({\overline{Q}}_c\right)\cap C_{1,2}\left(Q_c\right)$, $u_e\in C\left({\overline{Q}}_e\right)\cap C_{1,2}\left(Q_e\right)$ with initial conditions u(0, x),u_e(0, x) > 0, 0 ≤ b(0, x) ≤ b⁰. Then for t ∈ [0, T], 0 ≤ b ≤ b⁰, u, u_e are positive and bounded by a constant M which does not depend on u, b, u_e.

Proof.

Equation (2) can be written as $L_{b}b+K_b^{-}b+K_b^{+}bu=K_b^{-}{b}^0$, the first observation from it and (7) is that, by maximum principle (see Theorem 1.4–1.5, Chapter 2 in [27]), and as long as u ≥ 0, we have b ≥ 0. Next, since b⁰ is the solution of $L_{b}b=K_b^{-}\left(b^0-b\right)$, by comparison theorem, see [26, 27], we get b ≤ b⁰. The initial values of u, u_e are positive, so if u, u_e are not always positive on ${\overline{Q}}_c$, then suppose one of them, e.g., u becomes 0 no later than u_e and define $\underset{\_}{t}=\min \left\{\tau \ge 0\mid u(\tau ,\underset{\_}{x})=0,\underset{\_}{x}\in {\overline{\mathrm{\Omega }}}_c\right\}$ so that $u(\underset{\_}{t},\underset{\_}{x})=0$. Equation (1) can be written as $L_{c}u+K_b^{+}bu=K_b^{-}\left(b^0-b\right)$, where 0 ≤ b ≤ b⁰ on $[0,\underset{\_}{t}]$, by maximum principle and g_c(0) ≥ 0, −g_e(0, u_e) ≥ 0, we conclude that $\underset{\_}{t}$ does not exist. The positivity of u_e can be obtained similarly.

As u, u_e, b are non-negative and b is bounded, we can see that u is bounded by u_e. The following equation (30) is used to get the bound of u:

$$\{\begin{array}{l}{L}_{c}w=K_b^{-}{b}^0,\hfill \\ B_{c}w=C_3{c}_o/D_c,B_{e}w=\left(C_1^e+C_3^e\right)u_e/D_c,\hfill \\ w(0,x)=0.\hfill \end{array}$$ (30)

We define a constant w₀ > u(0, x) for $x\in {\overline{\mathrm{\Omega }}}_c$, then by comparison theorem, we have u(t, x) ≤ w(t, x) + w₀ on ${\overline{\mathrm{\Omega }}}_c\times [0,T]$. Just like (25) in Section 3.1, the solution w can be obtained as

$$w(t,x)={\int }_0^t{\int }_{\partial {\mathrm{\Omega }}_c}{\mathrm{\Gamma }}_c(t,x;\tau ,\xi ){\psi }_w(\tau ,\xi )d\xi d\tau +K_b^{-}{b}^0{\int }_0^t{\int }_{{\mathrm{\Omega }}_c}{\mathrm{\Gamma }}_c(t,x;\tau ,\xi )d\xi d\tau ,$$

$${\psi }_w(t,x)=2{\int }_0^t{\int }_{\partial {\mathrm{\Omega }}_c}{R}_c(t,x;\tau ,\xi )H_w(\tau ,\xi )d\xi d\tau +2H_w(t,x),$$

$$H_w(t,x)=K_b^{-}{b}^0{\int }_0^t{\int }_{{\mathrm{\Omega }}_c}\frac{\partial {\mathrm{\Gamma }}_c(t,x;\tau ,\xi )}{\partial v(t,x)}d\xi d\tau +g_w(t,x),$$

where g_w(t, x) = B_cw on ∂Ω and g_w(t, x) = B_ew on Υ. Let t ≤ h, h > 0, and define the norm ${\Vert u_e\Vert }_h=\sup \left\{\left|u_e(t,x)\right|;0\le t\le h,x\in {\overline{\mathrm{\Omega }}}_e\right\}$, by the properties of Γ_c as in (23)–(24), R_c as in (28)–(29), we have

$$\left|H_w(t,x)\right|\le C_1^w{t}^{1-\mu }+\left(C_1^e+C_3^e\right){\Vert u_e\Vert }_h/D_c+C_3{c}_o/D_c,$$

$$\left|{\psi }_w(t,x)\right|\le C_2^w\left(t^{1-\mu }+t^{2-2\mu }+c_o\right)+C_3^w\left(1+t^{1-\mu }\right){\Vert u_e\Vert }_h,$$

$$|w(t,x)|\le C_4^w\left(t^{1-\mu }+t^{2-2\mu }+t^{3-3\mu }\right)+C_5^w\left(t^{1-\mu }+t^{2-2\mu }\right){\Vert u_e\Vert }_h,$$

so that on ${\overline{\mathrm{\Omega }}}_c\times [0,h]$, u can be bounded as

$$|u(t,x)|\le C_4^w\left(t^{1-\mu }+t^{2-2\mu }+t^{3-3\mu }\right)+C_5^w\left(t^{1-\mu }+t^{2-2\mu }\right){\Vert u_e\Vert }_h+w_0.$$ (31)

Here, $C_1^w$ to $C_5^w$ do not depend on (t, x) and u_e.

We then prove that u_e is bounded by u from the following equation (32):

$$\{\begin{array}{l}{L}_{e}v=0,\hfill \\ {\partial }_{n}v=\left(C_1^e+2C_2^e/\left(K_{s}m\right)+C_3^e\right)u/D_e,\hfill \\ v(0,x)=0,\hfill \end{array}$$ (32)

where we define v₀ as a constant and v₀ > u_e(0, x) for $x\in {\overline{\mathrm{\Omega }}}_e$. By the comparison theorem, u_e(t, x) ≤ v(t,x) + v₀ on ${\overline{\mathrm{\Omega }}}_e\times [0,T]$. The solution v can be obtained as

$$v(t,x)={\int }_0^t{\int }_{{\rm Y}}{\mathrm{\Gamma }}_e(t,x;\tau ,\xi ){\psi }_v(\tau ,\xi )d\xi d\tau ,$$

$${\psi }_v(t,x)=2{\int }_0^t{\int }_{{\rm Y}}{R}_e(t,x;\tau ,\xi )g_v(\tau ,\xi )d\xi d\tau +2g_v(t,x),$$

where g_v(t, x) = ∂_nv. When defining the norm $\Vert u{\Vert }_h=\sup \{|u(t,x)|;0\le t\le h,x\in {\overline{\mathrm{\Omega }}}_c\}$, and by the properties of Γ_e(t, x; τ, ξ) from (23)–(24), R_e from (28)–(29), we get

$$\left|{\psi }_v(t,x)\right|\le C_1^v\left(1+t^{1-\mu }\right)\Vert u{\Vert }_h,|v(t,x)|\le C_2^v\left(t^{1-\mu }+t^{2-2\mu }\right)\Vert u{\Vert }_h,$$

so that on ${\overline{\mathrm{\Omega }}}_e\times [0,h]$, u_e can be bounded by

$$\left|u_e(t,x)\right|\le C_2^v\left(t^{1-\mu }+t^{2-2\mu }\right)\Vert u{\Vert }_h+v_0,$$ (33)

where $C_1^v$, $C_2^v$ do not depend on (t, x) and u. With (31), (33), and h small enough, we have $\Vert u{\Vert }_h\le \frac{1}{2}{\Vert u_e\Vert }_h+w_0+\frac{3}{4}{C}_4^w$, ${\Vert u_e\Vert }_h\le \frac{1}{2}\Vert u{\Vert }_h+v_0$, so that

$$\Vert u{\Vert }_h\le \frac{2}{3}{v}_0+\frac{4}{3}{w}_0+C_4^w,{\Vert u_e\Vert }_h\le \frac{4}{3}{v}_0+\frac{2}{3}{w}_0+\frac{1}{2}{C}_4^w.$$

Then, on time interval [h, 2h], let $w_1=\frac{2}{3}{v}_0+\frac{4}{3}{w}_0+C_4^w$, $v_1=\frac{4}{3}{v}_0+\frac{2}{3}{w}_0+\frac{1}{2}{C}_4^w$, by (30) and (32), we have u(t, x) ≤ w(t, x) + w₁ on ${\overline{\mathrm{\Omega }}}_c\times [h,2h]$ and u_e(t, x) ≤ v(t, x) + v₁ on ${\overline{\mathrm{\Omega }}}_e\times [h,2h]$. With the same coefficients as in the previous step, we get similar bounds of u, u_e for t ∈ [h, 2h]. Since h is fixed, with finite steps, we can reach the conclusion that u and u_e are bounded for t ∈ [0, T]. □

Theorem 3.2.

Suppose u(0, x),u_e(0, x) > 0 and b⁰ ≥ b(0, x) ≥ 0 are continuously differentiable in Ω_c or Ω_e, then there exists a unique solution {u, b, u_e} for (1)–(9).

Proof.

From Theorem 3.1, we can get the bounds of u,u_e and b if the solution {u, b, u_e} exits and its components are smooth. Let M be the upper bound of those three. We define ϕ(x) = max(0, min(x, M)), but ϕ can be smoother if needed, see Section 4.1. Then we change u, b to be ϕ(u), ϕ(b) in f(b, u) for equations (1)–(2), and replace u, u_e by ϕ(u), ϕ(u_e) in the right hand sides of (4)–(6). We assert that the modified problem has a unique solution and the solution has the same bounds. Thus, it is the same solution of the original system (1)–(9). Define the map T as {u, b, u_e} = T{w, w_b, w_e} for the modified problem

$$\{\begin{array}{l}{L}_{c}u=f\left(\varphi \left(w_b\right),\varphi (w)\right),\hfill \\ L_{b}b=f\left(\varphi \left(w_b\right),\varphi (w)\right),\hfill \\ L_e{u}_e=0,\hfill \\ B_{c}u=g_c(\varphi (w))/D_c,B_{e}u=-g_e\left(\varphi (w),\varphi \left(w_e\right)\right)/D_c,\hfill \\ {\partial }_n{u}_e=g_e\left(\varphi (w),\varphi \left(w_e\right)\right)/D_e,{\partial }_{n}b=0,\hfill \end{array}$$ (34)

where {w, w_b, w_e} are given functions, {u, b, u_e} is the solution of (34). Next we show T is a contraction map if t ≤ h for h small enough. To prove this, let the entries of {w₁, w_b,1, w_e,1}, {w₂, w_b,2, w_e,2} be Hölder continuous functions and {u₁, b₁, u_e,1} = T{w₁, w_b,1, w_e,1}, {u₂, b₂, u_e,2} = T{w₂, w_b,2, w_e,2}. Then we define the norms

$$\Vert v{\Vert }_h=\sup \left\{\right|v(t,x)|;0\le t\le h,x\in \overline{D}\},{\Vert \left\{v,v_b,v_e\right\}\Vert }_h=\Vert v{\Vert }_h+{\Vert v_b\Vert }_h+{\Vert v_e\Vert }_h,$$

where $\overline{D}$ can be ${\overline{\mathrm{\Omega }}}_c$ or ${\overline{\mathrm{\Omega }}}_e$. Let v = u₁ − u₂, v_b = b₁ − b₂, v_e = u_e,1 − u_e,2, and q = w₁ − w₂, q_b = w_b,1 − w_b,2, q_e = w_e,1 − w_e,2. Since the system (34) is fully decoupled, v, v_b, v_e can be solved separately. We start with v in (35)

$$\{\begin{array}{l}{L}_{c}v=-K_b^{+}{q}_b-K_b^{+}\left(\varphi \left(w_{b,1}\right)\varphi \left(w_1\right)-\varphi \left(w_{b,2}\right)\varphi \left(w_2\right)\right),\hfill \\ B_{c}v=\left(g_c\left(\varphi \left(w_1\right)\right)-g_c\left(\varphi \left(w_2\right)\right)\right)/D_c,\hfill \\ B_{e}v=\left(g_e\left(\varphi \left(w_1\right),\varphi \left(w_{e,1}\right)\right)-g_e\left(\varphi \left(w_2\right),\varphi \left(w_{e,2}\right)\right)\right)/D_c,\hfill \\ v(0,x)=0,\hfill \end{array}$$ (35)

and by Lemma 2.2, we have

$$|g_c\left(\varphi \left(w_1\right)\right)-g_c\left(\varphi \left(w_2\right)\right)|\le K_1|q|,$$

$$\left|g_e\left(\varphi \left(w_1\right),\varphi \left(w_{e,1}\right)\right)-g_e\left(\varphi \left(w_2\right),\varphi \left(w_{e,2}\right)\right)\right|\le K_2\left(|q|+\left|q_e\right|+{\int }_0^t|q(s)|\text{d}s\right).$$

Further, let $f_v(t,x)=-K_b^{+}{q}_b-K_b^{+}\left(\varphi \left(w_{b,1}\right)\varphi \left(w_1\right)-\varphi \left(w_{b,2}\right)\varphi \left(w_2\right)\right)$, g(t, x) = B_cv on ∂Ω and g(t, x) = B_ev on Υ. The solution of (35) is

$$v(t,x)={\int }_0^t{\int }_{\partial {\mathrm{\Omega }}_c}{\mathrm{\Gamma }}_c(t,x;\tau ,\xi ){\psi }_v(\tau ,\xi )d\xi d\tau +{\int }_0^t{\int }_{{\mathrm{\Omega }}_c}{\mathrm{\Gamma }}_c(t,x;\tau ,\xi )f_v(\tau ,\xi )d\xi d\tau ,$$

$${\psi }_v(t,x)=2H_v(t,x)+2{\int }_0^t{\int }_{\partial {\mathrm{\Omega }}_c}{R}_c(t,x;\tau ,\xi )H_v(\tau ,\xi )d\xi d\tau ,$$

$$H_v(t,x)={\int }_0^t{\int }_{{\mathrm{\Omega }}_c}\frac{\partial {\mathrm{\Gamma }}_c(t,x;\tau ,\xi )}{\partial v(t,x)}{f}_v(\tau ,\xi )d\xi d\tau +g(t,x).$$

By the properties of Γ_c, R_c, and t ≤ h, we have

$$\left|H_v(t,x)\right|\le C_1^v\left(t^{1-\mu }+1+t\right)\left(\Vert q{\Vert }_h+{\Vert q_b\Vert }_h+{\Vert q_e\Vert }_h\right),$$

$$\left|{\psi }_v(t,x)\right|\le C_2^v\left(t^{1-\mu }+1+t+t^{2-\mu }+t^{2-2\mu }\right)\left(\Vert q{\Vert }_h+{\Vert q_b\Vert }_h+{\Vert q_e\Vert }_h\right),$$

$$|v(t,x)|\le C_3^v\left(t^{1-\mu }+t^{2-\mu }+t^{2-2\mu }+t^{3-2\mu }+t^{3-3\mu }\right)\left(\Vert q{\Vert }_h+{\Vert q_b\Vert }_h+{\Vert q_e\Vert }_h\right),$$

where $C_1^v$, $C_2^v$, $C_3^v$ do not depend on q, q_b, q_e or (t, x). So we can choose h small enough such that $\Vert v{\Vert }_h\le 1/6\left(\Vert q{\Vert }_h+{\Vert q_b\Vert }_h+{\Vert q_e\Vert }_h\right)$. Similarly, we can obtain the bounds of v_b, v_e as ${\Vert v_b\Vert }_h\le 1/6\left(\Vert q{\Vert }_h+{\Vert q_b\Vert }_h\right)$, ${\Vert v_e\Vert }_h\le 1/6\left(\Vert q{\Vert }_h+{\Vert q_e\Vert }_h\right)$. Summing these terms we can show that T is a contraction map

$${\Vert T\left\{w_1,w_{b,1},w_{e,1}\right\}-T\left\{w_2,w_{b,2},w_{e,2}\right\}\Vert }_h\le \frac{1}{2}{\Vert \left\{w_1,w_{b,1},w_{e,1}\right\}-\left\{w_2,w_{b,2},w_{e,2}\right\}\Vert }_h,$$

where t ∈ [0, h]. By iteration, we can get the unique solution {u, b, u_e} on [0, h]. Then, choosing u(h, x),b(h, x),u_e(h, x) as the initial value, and repeating the steps above, the existence and uniqueness of the solution on [0, T] can be obtained. Following the proof in Theorem 3.1, it’s easy to see the bounds of the solution for the modified system are the same as the original system. For regularity of the solution, we refer to [26] for more information. □

4. Galerkin Projection and the Error Estimates

In this section, we define a nonlinear problem (36) in variational form, which is used in Section 4.2 to define the Galerkin projection of u, u_e. The wellposedness of (36) is proved In Section 4.1. From Section 3.2, we get the bounds of the exact solution {u, b, u_e} for (1)–(9), the bounds are used to define the function ϕ(·), such that ϕ(u) = u, ϕ(b) = b and ϕ(u_e) = u_e when u, b, u_e are within the bounds. Since b is not coupled with u_e, we can use the normal Galerkin projection for it, which will be given later. We now replace u in J_R, J_S, J_N by ϕ(u) and u_e in J_R by ϕ(u_e), see Section 4.1 for a smoother φ. For brevity, we denote ${\overline{g}}_c(u)$ and ${\overline{g}}_e\left(u,u_e\right)$ as the modified nonlinear boundary conditions. Then the variational form is given by

$$\{\begin{array}{l}{a}_c(u,v)+\lambda (u,v)=<{\overline{g}}_c(u),v{>}_{\partial \mathrm{\Omega }}-<{\overline{g}}_e\left(u,u_e\right),v{>}_{\mathrm{\Upsilon }},\hfill \\ a_e\left(u_e,v_e\right)+\lambda \left(u_e,v_e\right)=<{\overline{g}}_e\left(u,u_e\right),v_e{>}_{\mathrm{\Upsilon }},\hfill \end{array}$$ (36)

where $a_c(u,v)={\left(D_c\nabla u,\nabla v\right)}_{{\mathrm{\Omega }}_c},a_e\left(u_e,v_e\right)={\left(D_e\nabla u_e,\nabla v_e\right)}_{{\mathrm{\Omega }}_e}$. The problem is to find solutions u ∈ L^∞(0,T;H¹(Ω_c)), u_e ∈ L^∞(0,T;H¹(Ω_e)), for any v ∈ H¹(Ω_c), v_e ∈ H¹(Ω_e) and λ > 0 large enough.

4.1. Wellposedness of the Variational Problem

The existence and uniqueness for the solution of (36) are proved in this section, well-posedness of the Galerkin Projection (46) can be obtained similarly.

We define a smoother function ϕ(·) which is used in the variational problem

$$\varphi (x)=\{\begin{array}{ll}-a,\hfill & \text{if}x\le -a\hfill \\ \frac{3x^5}{a^4}+\frac{7x^4}{a^3}+\frac{4x^3}{a^2}+x,\hfill & \text{if}-a<x<0\hfill \\ x,\hfill & \text{if}0\le x\le M\hfill \\ \frac{3{(x-M)}^5}{a^4}-\frac{7{(x-M)}^4}{a^3}+\frac{4{(x-M)}^3}{a^2}+x,\hfill & \text{if}M<x\le M+a\hfill \\ M+a,\hfill & \text{if}x>M+a\hfill \end{array}$$

where a > 0, M > 0, a is small enough.

Lemma 4.1.

The nonlinear problem (36) has a unique solution, provided the modified conditions (4)–(6).

Proof.

The method we use here is adapted from Appendix in [28]. We try to get the exact solution by iteration, let $u^0\in H^1\left({\overline{Q}}_c\right)$, $u_e^0\in H^1\left({\overline{Q}}_e\right)$ be given functions and n ≥ 0, then we have

$$a_c\left(u^{n+1},v\right)+\lambda \left(u^{n+1},v\right)=<-{\overline{g}}_e\left(u^n,u_e^n\right),v{>}_{\mathrm{\Upsilon }}+<{\overline{g}}_c\left(u^n\right),v{>}_{\partial \mathrm{\Omega }},$$ (37)

$$a_e\left(u_e^{n+1},v_e\right)+\lambda \left(u_e^{n+1},v_e\right)=<{\overline{g}}_e\left(u^n,u_e^n\right),v_e{>}_{\mathrm{\Upsilon }},$$ (38)

where $u^{n+1},u_e^{n+1}$ share same initial values as $u^0,u_e^0$.

Let $r^{n+1}=u^{n+1}-u^n,r_e^{n+1}=u_e^{n+1}-u_e^n$ and n ≥ 1, we have

$$\begin{gathered} D_c\left(\nabla r^{n+1},\nabla v\right)+\lambda \left(r^{n+1},v\right)=<-{\overline{g}}_e\left(u^n,u_e^n\right)+{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),v{>}_{\mathrm{\Upsilon }} \\ +<{\overline{g}}_c\left(u^n\right)-{\overline{g}}_c\left(u^{n-1}\right),v{>}_{\partial \mathrm{\Omega }}, \end{gathered}$$

$$D_e\left(\nabla r_e^{n+1},\nabla v_e\right)+\lambda \left(r_e^{n+1},v_e\right)=<{\overline{g}}_e\left(u^n,u_e^n\right)-{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),v_e{>}_{\mathrm{\Upsilon }},$$

then let v = rⁿ⁺¹ and $v_e=r_e^{n+1}$, we obtain

$$\begin{gathered} D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2=<-{\overline{g}}_e\left(u^n,u_e^n\right)+{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),r^{n+1}{>}_{\mathrm{\Upsilon }} \\ +<{\overline{g}}_c\left(u^n\right)-{\overline{g}}_c\left(u^{n-1}\right),r^{n+1}{>}_{\partial \mathrm{\Omega }}, \end{gathered}$$

$$D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2=<{\overline{g}}_e\left(u^n,u_e^n\right)-{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),r_e^{n+1}{>}_{\mathrm{\Upsilon }}.$$

Sum those two equations, we get (39)

$$\begin{gathered} D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2+D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2 \\ =<{\overline{g}}_c\left(u^n\right)-{\overline{g}}_c\left(u^{n-1}\right),r^{n+1}{>}_{\partial \mathrm{\Omega }} \\ +<{\overline{g}}_e\left(u^n,u_e^n\right)-{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),r_e^{n+1}-r^{n+1}{>}_{\mathrm{\Upsilon }}. \end{gathered}$$ (39)

By Lemma 2.2, the first term of the right hand side is bounded:

$$<{\overline{g}}_c\left(u^n\right)-{\overline{g}}_c\left(u^{n-1}\right),r^{n+1}{>}_{\partial \mathrm{\Omega }}\le \frac{K_1}{2}{\Vert r^n\Vert }_{L^2(\partial \mathrm{\Omega })}^2+\frac{K_1}{2}{\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2,$$ (40)

and the second term of the right hand side is bounded:

$$\begin{gathered} <{\overline{g}}_e\left(u^n,u_e^n\right)-{\overline{g}}_e\left(u^{n-1},u_e^{n-1}\right),r_e^{n+1}-r^{n+1}{>}_{\mathrm{\Upsilon }} \\ \le K_2<\left|r^n\right|+\left|r_e^n\right|+{\int }_0^t\left|r^n(s)\right|\text{d}s,r_e^{n+1}-r^{n+1}{>}_{\mathrm{\Upsilon }} \\ \le \frac{K_2}{2}{\Vert r^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+\frac{K_2}{2}{\Vert r_e^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+\frac{K_2}{2}{\int }_0^t{\Vert r^n(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s \\ +K_2(2+T)\left({\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right). \end{gathered}$$ (41)

By equations (39) to (41), we have

$$D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2+D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2$$

$$\le \frac{K_1}{2}{\Vert r^n\Vert }_{L^2(\partial \mathrm{\Omega })}^2+\frac{K_2}{2}{\Vert r^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+\frac{K_2}{2}{\Vert r_e^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+\frac{K_2}{2}{\int }_0^t{\Vert r^n(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s$$

$$+\frac{K_1}{2}{\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+K_2(2+T)\left({\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right),$$

take L^∞ norm of ${\Vert r^n\Vert }_{L^2(\partial \mathrm{\Omega })}^2$, ${\Vert r^n\Vert }_{L^2(\mathrm{\Upsilon })}^2$, ${\Vert r_e^n\Vert }_{L^2(\mathrm{\Upsilon })}^2$, we obtain (42)

$$\begin{gathered} D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2+D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2 \\ \le C_1\left({\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2\right) \\ +C_2\left({\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right). \end{gathered}$$ (42)

Let α > 1, add the same term to both sides of (42), we have (43)

$$\begin{gathered} D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2+D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2 \\ +\alpha C_1\left({\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right) \\ \le C_1\left({\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2\right) \\ +\left(\alpha C_1+C_2\right)\left({\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right). \end{gathered}$$ (43)

From Lemma 2.3 and λ is large enough, we have

$$\begin{gathered} \left(\alpha C_1+C_2\right)\left({\Vert r^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert r^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert r_e^{n+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right) \\ \le D_c{\Vert \nabla r^{n+1}\Vert }^2+\lambda {\Vert r^{n+1}\Vert }^2+D_e{\Vert \nabla r_e^{n+1}\Vert }^2+\lambda {\Vert r_e^{n+1}\Vert }^2. \end{gathered}$$ (44)

So that combine (43) and (44), we obtain (45)

$$\begin{gathered} {\Vert r^{n+1}\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^{n+1}\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^{n+1}\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2 \\ \le \frac{1}{\alpha }\left({\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2\right). \end{gathered}$$ (45)

From (42) and (45), let $\tau =\min \left\{D_c,D_e\right\},C_{\tau }=\frac{C_1+C_2/\alpha }{\tau }$, we arrive at

$${\Vert r^{n+1}\Vert }_{L^{\infty }\left(0,T;H^1\left({\mathrm{\Omega }}_c\right)\right)}^2+{\Vert r_e^{n+1}\Vert }_{L^{\infty }\left(0,T;H^1\left({\mathrm{\Omega }}_e\right)\right)}^2$$

$$\le C_{\tau }\left({\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^n\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2\right)$$

$$\le \frac{C_{\tau }}{{\alpha }^n}\left({\Vert r^1\Vert }_{L^{\infty }\left(0,T;L^2(\partial \mathrm{\Omega })\right)}^2+{\Vert r^1\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2+{\Vert r_e^1\Vert }_{L^{\infty }\left(0,T;L^2(\mathrm{\Upsilon })\right)}^2\right).$$

which implies that $u^n,u_e^n$ converge to a unique solution. □

4.2. Galerkin Projection

We define S_k(Ω_c), S_k(Ω_e), k ≥ 1 as the classical conforming P_k finite element spaces, with shape regular triangular/tetrahedral meshes and the size of the corresponding mesh is h. For curved boundaries, we refer to isoparametric finite element methods, see [29, 30] and references therein. The Galerkin projection for the exact solutions u, u_e can be obtained from (46):

$$\{\begin{array}{c}(a_c(u-W,v)+\lambda (u-W,v)=<-{\overline{g}}_e\left(u,u_e\right)+{\overline{g}}_e\left(W,W_e\right),v{>}_{\mathrm{\Upsilon }}\hfill \\ +<{\overline{g}}_c(u)-{\overline{g}}_c(W),v{>}_{\partial \mathrm{\Omega }},\\ a_e\left(u_e-W_e,v_e\right)+\lambda \left(u_e-W_e,v_e\right)=<{\overline{g}}_e\left(u,u_e\right)-{\overline{g}}_e\left(W,W_e\right),v_e{>}_{\mathrm{\Upsilon }}.\hfill \end{array}$$ (46)

where W(t), v ∈ S_k(Ω_c), and W_e(t), v_e ∈ S_k(Ω_c), λ is large enough.

Also without further notice, η, η_e are defined as $\eta :=u-W$, ${\eta }_e:=u_e-W_e$, the following norms are denoted as $\Vert \cdot {\Vert }_{\infty ;k}:=\Vert \cdot {\Vert }_{L^{\infty }\left(0,T;H^k(D)\right)}$, $\Vert \cdot {\Vert }_{\infty }{}_{;0}:=\Vert \cdot {\Vert }_{L^{\infty }\left(0,T;L^2(D)\right)}$, and $\Vert \cdot \Vert :=\Vert \cdot {\Vert }_{L^2(D)}$, where D can be Ω_c or Ω_e depending on the domain of the given function. Then we have the following theorem:

Theorem 4.1.

With P_k elements, k ≥ 1, and appropriately chosen W(0), W_e(0), we have

$$\Vert \eta {\Vert }_{\infty ;1}+{\Vert {\eta }_e\Vert }_{\infty ;1}\le C\left(\Vert u{\Vert }_{\infty ;k+1}+{\Vert u_e\Vert }_{\infty ;k+1}\right)h^k,$$ (47)

$$\begin{gathered} {\Vert {\partial }_t\eta \Vert }_{\infty ;1}+{\Vert {\partial }_t{\eta }_e\Vert }_{\infty ;1}\le C\left(\Vert u{\Vert }_{\infty ;k+1}+{\Vert {\partial }_{t}u\Vert }_{\infty ;k+1}\right)h^k \\ +C\left({\Vert u_e\Vert }_{\infty ;k+1}+{\Vert {\partial }_t{u}_e\Vert }_{\infty ;k+1}\right)h^k, \end{gathered}$$ (48)

where h is the mesh size and C does not depend on h.

Proof.

First step, we prove (47). Letting δu = w − u, δu_e = w_e − u_e, w ∈ S_k(Ω_c), w_e ∈ S_k(Ω_e), (46) can be written as

$$\{\begin{array}{c}{a}_c(\eta ,v)+\lambda (\eta ,v)=<-{\overline{g}}_e\left(u,u_e\right)+{\overline{g}}_e\left(W,W_e\right),v{>}_{\mathrm{\Upsilon }}\hfill \\ +<{\overline{g}}_c(u)-{\overline{g}}_c(W),v{>}_{\partial \mathrm{\Omega }},\\ a_e\left({\eta }_e,v_e\right)+\lambda \left({\eta }_e,v_e\right)=<{\overline{g}}_e\left(u,u_e\right)-{\overline{g}}_e\left(W,W_e\right),v_e{>}_{\mathrm{\Upsilon }}.\hfill \end{array}$$ (49)

Now set v = w − W = δu + η, v_e = w_e − W_e = δu_e + η_e then

$$\{\begin{array}{c}(a_c(\eta ,\delta u+\eta )+\lambda (\eta ,\delta u+\eta )=<-{\overline{g}}_e\left(u,u_e\right)+{\overline{g}}_e\left(W,W_e\right),\delta u+\eta {>}_{\mathrm{\Upsilon }}\hfill \\ +<{\overline{g}}_c(u)-{\overline{g}}_c(W),\delta u+\eta {>}_{\partial \mathrm{\Omega }},\\ a_e\left({\eta }_e,\delta u_e+{\eta }_e\right)+\lambda \left({\eta }_e,\delta u_e+{\eta }_e\right)=<{\overline{g}}_e\left(u,u_e\right)-{\overline{g}}_e\left(W,W_e\right),\delta u_e+{\eta }_e{>}_{\mathrm{\Upsilon }},\hfill \end{array}$$

moving the terms containing δu to the right hand side leads to (with Schwarz inequality, Lemma 2.2 and trace theorem):

$$D_c\Vert \nabla \eta {\Vert }^2+\lambda \Vert \eta {\Vert }^2+D_e{\Vert \nabla {\eta }_e\Vert }^2+\lambda {\Vert {\eta }_e\Vert }^2$$

$$\le \frac{D_c}{4}\Vert \nabla \eta {\Vert }^2+\frac{\lambda }{4}\Vert \eta {\Vert }^2+\frac{D_e}{4}{\Vert \nabla {\eta }_e\Vert }^2+\frac{\lambda }{4}{\Vert {\eta }_e\Vert }^2$$

$$+C_1\left(\Vert \eta {\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\eta }_e\Vert }_{L^2(\mathrm{\Upsilon })}^2+\Vert \eta {\Vert }_{L^2(\partial \mathrm{\Omega })}^2\right)+C_2{\int }_0^t\Vert \eta (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s$$

$$+C_3(\Vert \eta (0){\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\eta }_e(0)\Vert }_{L^2(\mathrm{\Upsilon })}^2+\underset{w\in S_k\left({\mathrm{\Omega }}_c\right)}{\inf }\Vert \delta u{\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2$$

$$+\underset{w_e\in S_k\left({\mathrm{\Omega }}_e\right)}{\inf }{\Vert \delta u_e\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2),$$

from Lemma 2.3, we have

$$C_2{\int }_0^t\Vert \eta (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\le {\int }_0^t{D}_c/(4T)\Vert \nabla \eta (s){\Vert }^2+C\Vert \eta (s){\Vert }^2\text{d}s,$$

the bounds for $C_1\left(\Vert \eta {\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\eta }_e\Vert }_{L^2(\mathrm{\Upsilon })}^2+\Vert \eta {\Vert }_{L^2(\partial \mathrm{\Omega })}^2\right)$ can be obtained similarly. Taking L^∞ norm with respect to t then to s for the right-hand side, then taking the L^∞ norm with respect to t for the left hand side, cancelling the corresponding terms, we obtain

$$\begin{gathered} \Vert \eta {\Vert }_{\infty ;1}^2+{\Vert {\eta }_e\Vert }_{\infty ;1}^2\le C(\Vert \eta (0){\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+{\Vert {\eta }_e(0)\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2 \\ +\underset{t\in [0,T]}{\sup }\underset{w\in S_k\left({\mathrm{\Omega }}_c\right)}{\inf }\Vert \delta u{\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2 \\ +\underset{t\in [0,T]}{\sup }\underset{w_e\in S_k\left({\mathrm{\Omega }}_e\right)}{\inf }{\Vert \delta u_e\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2), \end{gathered}$$

so that (47) is proved.

Next step, we prove (48). Differentiating ${\overline{g}}_c$ with respect to (w.r.t.) t produces

$$\frac{\text{d}{\overline{g}}_c(u)}{\text{d}t}=\frac{\partial {\overline{g}}_c}{\partial u}(u)\frac{\partial u}{\partial t},$$

$$\frac{\partial {\overline{g}}_c}{\partial u}(u)=-C_1^c\frac{2u{K}_p^2}{{\left(K_p^2+u^2\right)}^2}-C_2^c\frac{K_n}{{\left(K_n+\varphi (u)\right)}^2}\frac{\partial \varphi }{\partial u}-C_3^c.$$

The derivative of ${\overline{g}}_e\left(u,u_e\right)$ w.r.t. t is

$$\begin{gathered} \frac{\text{d}{\overline{g}}_e\left(u,u_e\right)}{\text{d}t}=C_1^e\left(c_1^{\prime }(t)+c_2^{\prime }(t)\right)\left(\varphi \left(u_e\right)-\varphi (u)\right) \\ -C_1^{e}P(t,\varphi (u))\left(\frac{\partial \varphi }{\partial u_e}\frac{\partial u_e}{\partial t}-\frac{\partial \varphi }{\partial u}\frac{\partial u}{\partial t}\right) \\ +\frac{\partial \psi }{\partial u}\frac{\partial u}{\partial t}+\frac{\partial \psi }{\partial u_e}\frac{\partial u_e}{\partial t}, \end{gathered}$$

where ψ = J_S(ϕ(u), u_e) + J_l,e, and

$$\frac{\partial \psi }{\partial u}=C_2^e\frac{K_s}{{\left(K_s+\varphi (u)\right)}^2{\varphi }_m\left(u_e\right)}\frac{\partial \varphi }{\partial u}-C_3^e,$$

$$\frac{\partial \psi }{\partial u_e}=-C_2^e\frac{\varphi (u)}{\left(K_s+\varphi (u)\right){\left({\varphi }_m\left(u_e\right)\right)}^2}\frac{\partial {\varphi }_m}{\partial u_e}+C_3^e.$$

Then (46) can be differentiated w.r.t. t as

$$\{\begin{array}{c}{a}_c\left(\frac{\partial \eta }{\partial t},v\right)+\lambda \left(\frac{\partial \eta }{\partial t},v\right)=<-\frac{\text{d}{\overline{g}}_e\left(u,u_e\right)}{\text{d}t}+\frac{\text{d}{\overline{g}}_e\left(W,W_e\right)}{\text{d}t},v{>}_{\mathrm{\Upsilon }}\hfill \\ +<\frac{\text{d}{\overline{g}}_c(u)}{\text{d}t}-\frac{\text{d}{\overline{g}}_c(W)}{\text{d}t},v{>}_{\partial \mathrm{\Omega }},\\ a_e\left(\frac{\partial {\eta }_e}{\partial t},v_e\right)+\lambda \left(\frac{\partial {\eta }_e}{\partial t},v_e\right)=<\frac{\text{d}{\overline{g}}_e\left(u,u_e\right)}{\text{d}t}-\frac{\text{d}{\overline{g}}_e\left(W,W_e\right)}{\text{d}t},v_e>\mathrm{\Upsilon }.\hfill \end{array}$$ (50)

All terms in the derivatives of ${\overline{g}}_c(u)$ and ${\overline{g}}_e\left(u,u_e\right)$ w.r.t. t, except $\frac{\text{d}}{\text{d}t}{J}_R\left(\varphi (u),\varphi \left(u_e\right)\right)$, are Lipschitz continuous. $\frac{\text{d}}{\text{d}t}{J}_R$ can be treated as in Lemma 2.1, which gives us

$$\begin{gathered} \left|\frac{\text{d}{\overline{g}}_c(u)}{\text{d}t}-\frac{\text{d}{\overline{g}}_c(W)}{\text{d}t}\right|\le C_1|\eta |\left|\frac{\partial u}{\partial t}\right|+\left|\frac{\partial {\overline{g}}_c}{\partial u}(W)\right|\left|\frac{\partial \eta }{\partial t}\right|, \\ \left|\frac{\text{d}{\overline{g}}_e\left(u,u_e\right)}{\text{d}t}-\frac{\text{d}{\overline{g}}_e\left(W,W_e\right)}{\text{d}t}\right|\le C_2\left(|\eta |+\left|{\eta }_e\right|+|\eta (0)|+{\int }_0^t|\eta (s)|\text{d}s\right) \\ +C_3\left(|\eta (0)|+{\int }_0^t|\eta (s)|\text{d}s\right)\left|\frac{\partial \varphi }{\partial u_e}\frac{\partial u_e}{\partial t}-\frac{\partial \varphi }{\partial u}\frac{\partial u}{\partial t}\right| \\ +C_3|\eta |\left|\frac{\partial u}{\partial t}\right|+C_3\left|\frac{\partial \varphi }{\partial u}(W)\right|\left|\frac{\partial \eta }{\partial t}\right| \\ +C_3\left|{\eta }_e\right|\left|\frac{\partial u_e}{\partial t}\right|+C_3\left|\frac{\partial \varphi }{\partial u_e}\left(W_e\right)\right|\left|\frac{\partial {\eta }_e}{\partial t}\right| \\ +C_4\left(|\eta |+\left|{\eta }_e\right|\right)\left|\frac{\partial u}{\partial t}\right|+\left|\frac{\partial \psi }{\partial u}\left(W,W_e\right)\right|\left|\frac{\partial \eta }{\partial t}\right| \\ +C_5\left(|\eta |+\left|{\eta }_e\right|\right)\left|\frac{\partial u_e}{\partial t}\right|+\left|\frac{\partial \psi }{\partial u_e}\left(W,W_e\right)\right|\left|\frac{\partial {\eta }_e}{\partial t}\right|, \end{gathered}$$

where $\frac{\partial \varphi }{\partial u}$, $\frac{\partial \varphi }{\partial u_e}$, $\frac{\partial \psi }{\partial u}$, $\frac{\partial \psi }{\partial u_e}$ are bounded from the definitions of ϕ(·) in Section 4.1 and ϕ_m(·) in Section 1. Similar as the estimates for η, η_e, (48) can be obtained. □

5. Error Estimate for the Semi-discrete Galerkin Method

Without further notice, in the sequel, we use the modified system (change f(b, u) to f(ϕ(b), ϕ(u)), g_c(u) to ${\overline{g}}_c(u)$ and g_e(u, u_e) to ${\overline{g}}_e\left(u,u_e\right)$ as in Section 4) to obtain the error analysis since it has the same solution as the original system, see a similar proof in Theorem 3.2. Letting U(t), U_b(t) ∈ S_k(Ω_c), U_e(t) ∈ S_k(Ω_e), for t ∈ [0, T] and v, v_b ∈ S_k(Ω_c), v_e ∈ S_k(Ω_e), the semi-discrete form of the model is given as

$$\{\begin{array}{c}\left({\partial }_{t}U,v\right)+a_c(U,v)=<g_c(U),v{>}_{\partial \mathrm{\Omega }}-<g_e\left(U,U_e\right),v{>}_{\mathrm{\Upsilon }}\hfill \\ +\left(f\left(U_b,U\right),v\right),\\ \left({\partial }_t{U}_b,v_b\right)+a_b\left(U_b,v_b\right)=\left(f\left(U_b,U\right),v_b\right),\hfill \\ \left({\partial }_t{U}_e,v_e\right)+a_e\left(U_e,v_e\right)=<g_e\left(U,U_e\right),v_e{>}_{\mathrm{\Upsilon }}.\hfill \end{array}$$ (51)

Then from the model equations (1)–(9) and the nonlinear projection (46), we have

$$\{\begin{array}{c}\left({\partial }_{t}W,v\right)+a_c(W,v)\hfill \\ =<g_c(W),v{>}_{\partial \mathrm{\Omega }}-<g_e\left(W,W_e\right),v{>}_{\mathrm{\Upsilon }}+\left(f\left(W_b,W\right),v\right)\\ +\lambda (u-W,v)-\left({\partial }_t\eta ,v\right)+\left(f(b,u)-f\left(W_b,W\right),v\right),\\ \left({\partial }_t{W}_b,v_b\right)+a_b\left(W_b,v_b\right)\hfill \\ =\left(f\left(W_b,W\right),v_b\right)\\ +\left(b-W_b,v_b\right)-\left({\partial }_t{\eta }_b,v_b\right)+\left(f(b,u)-f\left(W_b,W\right),v_b\right),\\ \left({\partial }_t{W}_e,v_e\right)+a_e\left(W_e,v_e\right)\hfill \\ =<g_e\left(W,W_e\right),v_e{>}_{\mathrm{\Upsilon }}+\lambda \left(u_e-W_e,v_e\right)-\left({\partial }_t{\eta }_e,v_e\right),\end{array}$$ (52)

where η = u − W, η_b = b − W_b, η_e = u_e − W_e, and W_b(t) ∈ S_k(Ω_c) is the Galerkin Projection of b from $\left(D_b\nabla \left(b-W_b\right),\nabla v_b\right)+\left(b-W_b,v_b\right)=0$. Letting e = u − U, e_b = b − U_b, e_e = u_e − U_e, we get the following theorem:

Theorem 5.1.

If the exact solutions u, b, u_e for (1)–(9) are smooth enough, then with P_k elements, k ≥ 1, and appropriately chosen U(0), U_e(0), U_b(0), we have

$$\begin{gathered} \Vert e{\Vert }_{\infty ;0}^2+{\Vert e_b\Vert }_{\infty ;0}^2+{\Vert e_e\Vert }_{\infty ;0}^2 \\ +{\int }_0^T\Vert \nabla e(s){\Vert }^2+{\Vert \nabla e_b(s)\Vert }^2+{\Vert \nabla e_e(s)\Vert }^2\text{d}s \\ \le C{h}^{2k}(\Vert u{\Vert }_{\infty ;k+1}^2+{\Vert {\partial }_{t}u\Vert }_{\infty ;k+1}^2+\Vert b{\Vert }_{\infty ;k+1}^2+{\Vert {\partial }_{t}b\Vert }_{\infty ;k+1}^2 \end{gathered}$$

$$+{\Vert u_e\Vert }_{\infty ;k+1}^2+{\Vert {\partial }_t{u}_e\Vert }_{\infty ;k+1}^2),$$

where h is the mesh size, C does not depend on h.

Proof.

Letting ξ = W − U, ξ_b = W_b − U_b, ξ_e = W_e − U_e and η = u − W, η_b = b − W_b, η_e = u_e − W_e, subtracting (51) from (52), we have

$$\{\begin{array}{c}\left({\partial }_t\xi ,v\right)+a_c(\xi ,v)\hfill \\ =<g_c(W)-g_c(U),v{>}_{\partial \mathrm{\Omega }}-<g_e\left(W,W_e\right)-g_e\left(U,U_e\right),v{>}_{\mathrm{\Upsilon }}\\ +\left(f\left(W_b,W\right)-f\left(U_b,U\right),v\right)\\ +\lambda (\eta ,v)-\left({\partial }_t\eta ,v\right)+\left(f(b,u)-f\left(W_b,W\right),v\right),\\ \left({\partial }_t{\xi }_b,v_b\right)+a_b\left({\xi }_b,v_b\right)\hfill \\ =\left(f\left(W_b,W\right)-f\left(U_b,U\right),v_b\right)+\left({\eta }_b,v_b\right)\\ -\left({\partial }_t{\eta }_b,v_b\right)+\left(f(b,u)-f\left(W_b,W\right),v_b\right),\\ \left({\partial }_t{\xi }_e,v_e\right)+a_e\left({\xi }_e,v_e\right)\hfill \\ =<g_e\left(W,W_e\right)-g_e\left(U,U_e\right),v_e>{\rm Y}+\lambda \left({\eta }_e,v_e\right)-\left({\partial }_t{\eta }_e,v_e\right).\end{array}$$ (53)

From Lemma 2.2 and letting W(0) = U(0), we have

$$\left|g_c(W)-g_c(U)\right|\le K_1\left|\xi \right|,$$ (54)

$$|g_e\left(W,W_e\right)-g_e\left(U,U_e\right)|\le K_2\left(|\xi |+|{\xi }_e|+{\int }_0^t|\xi (s)|\text{d}s\right),$$ (55)

$$\left|f\left(W_b,W\right)-f\left(U_b,U\right)\right|\le K_3\left(\left|{\xi }_b\right|+\left|{\xi }_e\right|\right).$$ (56)

Then in (53), let v = ξ, v_b = ξ_b, v_e = ξ_e, we obtain (57) – (59):

$$\begin{gathered} \frac{1}{2}\frac{\text{d}}{\text{d}t}\Vert \xi {\Vert }^2+D_c\Vert \nabla \xi {\Vert }^2 \\ \le C_1\left(\Vert \xi {\Vert }_{L^2(\partial \mathrm{\Omega })}^2+\Vert \xi {\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^t\Vert \xi (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_1\left({\Vert {\xi }_b\Vert }^2+\Vert \xi {\Vert }^2\right) \\ +\frac{\lambda +K_3}{2}\Vert \eta {\Vert }^2+\frac{1}{2}{\Vert {\partial }_t\eta \Vert }^2+\frac{K_3}{2}{\Vert {\eta }_b\Vert }^2, \end{gathered}$$ (57)

$$\begin{gathered} \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\xi }_b\Vert }^2+D_b{\Vert \nabla {\xi }_b\Vert }^2\le C_1\left({\Vert {\xi }_b\Vert }^2+\Vert \xi {\Vert }^2\right) \\ +\frac{1+K_3}{2}{\Vert {\eta }_b\Vert }^2+\frac{1}{2}{\Vert {\partial }_t{\eta }_b\Vert }^2+\frac{K_3}{2}\Vert \eta {\Vert }^2, \end{gathered}$$ (58)

$$\begin{gathered} \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\xi }_e\Vert }^2+D_e{\Vert \nabla {\xi }_e\Vert }^2 \\ \le C_3\left(\Vert \xi {\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^t\Vert \xi (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s+{\Vert {\xi }_e\Vert }^2\right) \\ +\frac{\lambda }{2}{\Vert {\eta }_e\Vert }^2+\frac{1}{2}{\Vert {\partial }_t{\eta }_e\Vert }^2. \end{gathered}$$ (59)

The sum of (57), (58) and (59) gives the following inequality:

$$\begin{gathered} \frac{1}{2}\frac{\text{d}}{\text{d}t}\left(\Vert \xi {\Vert }^2+{\Vert {\xi }_b\Vert }^2+{\Vert {\xi }_e\Vert }^2\right)+D_c\Vert \nabla \xi {\Vert }^2+D_b{\Vert \nabla {\xi }_b\Vert }^2+D_e{\Vert \nabla {\xi }_e\Vert }^2 \\ \le C_1\left(\Vert \xi {\Vert }_{L^2(\partial \mathrm{\Omega })}^2+\Vert \xi {\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^t\Vert \xi (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_2\left({\Vert {\xi }_b\Vert }^2+\Vert \xi {\Vert }^2+{\Vert {\xi }_e\Vert }^2\right) \\ +C_3(\Vert \eta {\Vert }^2+{\Vert {\partial }_t\eta \Vert }^2+{\Vert {\eta }_b\Vert }^2+{\Vert {\partial }_t{\eta }_b\Vert }^2 \\ +{\Vert {\eta }_e\Vert }^2+{\Vert {\partial }_t{\eta }_e\Vert }^2). \end{gathered}$$ (60)

By Lemma 2.3 and

$${\int }_0^t{\int }_0^s\Vert \xi (w){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}w\text{d}s\le {\int }_0^t{\int }_0^t\Vert \xi (w){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}w\text{d}s\le T{\int }_0^t\Vert \xi (s){\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s$$

from (60), we have

$$\begin{gathered} \Vert \xi (t){\Vert }^2+{\Vert {\xi }_b(t)\Vert }^2+{\Vert {\xi }_e(t)\Vert }^2+{\int }_0^t\Vert \nabla \xi (s){\Vert }^2+{\Vert \nabla {\xi }_b(s)\Vert }^2+{\Vert \nabla {\xi }_e(s)\Vert }^2\text{d}s \\ \le C_1{\int }_0^t{\Vert {\xi }_b(s)\Vert }^2+\Vert \xi (s){\Vert }^2+{\Vert {\xi }_e(s)\Vert }^2\text{d}s \\ +C_2{\int }_0^t\Vert \eta {\Vert }^2+{\Vert {\partial }_t\eta \Vert }^2+{\Vert {\eta }_b\Vert }^2+{\Vert {\partial }_t{\eta }_b\Vert }^2+{\Vert {\eta }_e\Vert }^2+{\Vert {\partial }_t{\eta }_e\Vert }^2\text{d}s. \end{gathered}$$ (61)

From (61), by Gronwall’s Lemma, the following estimate can be obtained:

$$\begin{gathered} \Vert \xi (t){\Vert }^2+{\Vert {\xi }_b(t)\Vert }^2+{\Vert {\xi }_e(t)\Vert }^2 \\ +{\int }_0^t\Vert \nabla \xi (s){\Vert }^2+{\Vert \nabla {\xi }_b(s)\Vert }^2+{\Vert \nabla {\xi }_e(s)\Vert }^2\text{d}s \\ \le C{\int }_0^t\Vert \eta {\Vert }^2+{\Vert {\partial }_t\eta \Vert }^2+{\Vert {\eta }_b\Vert }^2+{\Vert {\partial }_t{\eta }_b\Vert }^2+{\Vert {\eta }_e\Vert }^2+{\Vert {\partial }_t{\eta }_e\Vert }^2\text{d}s, \end{gathered}$$

then by ξ = e − η, ξ_b = e_b − η_b, ξ_e = e_e − η_e, we have

$$\begin{gathered} \Vert e{\Vert }_{\infty ;0}^2+{\Vert e_b\Vert }_{\infty ;0}^2+{\Vert e_e\Vert }_{\infty ;0}^2+{\int }_0^T\Vert \nabla e(s){\Vert }^2+{\Vert \nabla e_b(s)\Vert }^2+{\Vert \nabla e_e(s)\Vert }^2\text{d}s \\ \le C(\Vert \eta {\Vert }_{\infty ;1}^2+{\Vert {\partial }_t\eta \Vert }_{\infty ;1}^2+{\Vert {\eta }_b\Vert }_{\infty ;1}^2+{\Vert {\partial }_t{\eta }_b\Vert }_{\infty ;1}^2 \\ +{\Vert {\eta }_e\Vert }_{\infty ;1}^2+{\Vert {\partial }_t{\eta }_e\Vert }_{\infty ;1}^2) \\ +C\left(\Vert \eta (0){\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+{\Vert {\eta }_b(0)\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+{\Vert {\eta }_e(0)\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2\right), \end{gathered}$$ (62)

and with Theorem 4.1, the proof is completed. □

6. Error Estimate for the Fully Discrete Implicit-Explicit Scheme

We define d_tUⁿ⁺¹ = (Uⁿ⁺¹ − Uⁿ)/Δt, where Δt = T/N, N is a positive integer, n = 0, 1, 2, ⋯, N − 1, Uⁿ ∈ S_k(Ω_c). Also $d_t{U}_b^{n+1}$, $d_t{U}_e^{n+1}$ can be defined similarly, where $U_b^n\in S_k\left({\mathrm{\Omega }}_c\right)$, $U_e^n\in S_k\left({\mathrm{\Omega }}_e\right)$. Then let v, v_b ∈ S_k(Ω_c), v_e ∈ S_k(Ω_e) and t_n = nΔt, we have the fully discrete form of the model:

$$\{\begin{array}{c}\left(d_t{U}^{n+1},v\right)+a_c\left(U^{n+1},v\right)=<g_c\left(U^n\right),v{>}_{\partial \mathrm{\Omega }}-<g_e\left(U^n,U_e^n\right),v{>}_{\mathrm{\Upsilon }}\hfill \\ +\left(f\left(U_b^n,U^n\right),v\right)\\ \left(d_t{U}_b^{n+1},v_b\right)+a_b\left(U_b^{n+1},v_b\right)=\left(f\left(U_b^n,U^n\right),v_b\right)\hfill \\ \left(d_t{U}_e^{n+1},v_e\right)+a_e\left(U_e^{n+1},v_e\right)=<g_e\left(U^n,U_e^n\right),v_e{>}_{\mathrm{\Upsilon }},\hfill \end{array}$$ (63)

for the ODE part, we employ $U_h(t,x)=\sum _{i=0}^n{\varphi }_i(t)U^i(x)$ for 0 ≤ t ≤ t_n, where ϕ_i is the one-dimensional hat function and ϕ_i(t_i) = 1. Let eⁿ = u(t_n) − Uⁿ, $e_b^n=b\left(t_n\right)-U_b^n$, $e_e^n=u_e\left(t_n\right)-U_e^n$, where u, b, u_e are the exact solutions for (1)–(9), then we have the following theorem:

Theorem 6.1.

If the exact solutions u, b, u_e are sufficiently smooth, then with P_k elements, k ≥ 1, appropriately chosen U⁰, $U_b^0$, $U_e^0$, and sufficiently small Δt which doesn’t depend on the spatial mesh size, for any n ≤ N − 1, we have

$$\begin{gathered} {\Vert e^{n+1}\Vert }^2+{\Vert e_b^{n+1}\Vert }^2+{\Vert e_e^{n+1}\Vert }^2 \\ +\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla e^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla e_b^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla e_e^l\Vert }^2\le C\left(\mathrm{\Delta }{t}^2+h^{2k}\right), \end{gathered}$$

where Δt is the time-step size, h is the mesh size, C does not depend on n, h and Δt.

Proof.

Let uⁿ = u(t_n), d_tuⁿ⁺¹ = (uⁿ⁺¹ − uⁿ)/Δt and similarly we have d_tηⁿ⁺¹, $d_t{\eta }_b^{n+1}$, $d_t{\eta }_e^{n+1}$, where ${\eta }^n=u\left(t_n\right)-W\left(t_n\right),{\eta }_b^n=b\left(t_n\right)-W_b\left(t_n\right)$, ${\eta }_e^n=u_e\left(t_n\right)-W_e\left(t_n\right)$, W, W_b, W_e are the previously defined Galerkin projections for u, b, u_e.

Then we have

$$\{\begin{array}{c}\left(d_t{W}^{n+1},v\right)+a_c\left(W^{n+1},v\right)=<g_c\left(W^n\right),v{>}_{\partial \mathrm{\Omega }}\hfill \\ -<g_e\left(W^n,W_e^n\right),v{>}_{\mathrm{\Upsilon }}\\ +\left(f\left(W_b^n,W^n\right),v\right)+E^{n+1}(v),\\ \left(d_t{W}_b^{n+1},v_b\right)+a_b\left(W_b^{n+1},v_b\right)=\left(f\left(W_b^n,W^n\right),v_b\right)+E_b^{n+1}\left(v_b\right),\hfill \\ \left(d_t{W}_e^{n+1},v_e\right)+a_e\left(W_e^{n+1},v_e\right)=<g_e\left(W^n,W_e^n\right),v_e{>}_{{\rm Y}}+E_e^{n+1}\left(v_e\right),\hfill \end{array}$$ (64)

where

$$\begin{gathered} E^{n+1}(v)=<g_c\left(W^{n+1}\right)-g_c\left(W^n\right),v{>}_{\partial \mathrm{\Omega }} \\ -<g_e\left(W^{n+1},W_e^{n+1}\right)-g_e\left(W^n,W_e^n\right),v{>}_{\mathrm{\Upsilon }} \\ +\left(f\left(W_b^{n+1},W^{n+1}\right)-f\left(W_b^n,W^n\right),v\right)-\left(d_t{\eta }^{n+1},v\right) \\ +\left(d_t{u}^{n+1}-{\partial }_t{u}^{n+1},v\right)+\lambda \left({\eta }^{n+1},v\right) \\ +\left(f\left(b^{n+1},u^{n+1}\right)-f\left(W_b^{n+1},W^{n+1}\right),v\right), \end{gathered}$$

$$\begin{gathered} E_b^{n+1}\left(v_b\right)=\left(f\left(W_b^{n+1},W^{n+1}\right)-f\left(W_b^n,W^n\right),v_b\right)-\left(d_t{\eta }_b^{n+1},v_b\right) \\ +\left(d_t{b}^{n+1}-{\partial }_t{b}^{n+1},v_b\right)+\left({\eta }_b^{n+1},v_b\right) \\ +\left(f\left(b^{n+1},u^{n+1}\right)-f\left(W_b^{n+1},W^{n+1}\right),v_b\right), \end{gathered}$$

$$\begin{gathered} E_e^{n+1}\left(v_e\right)=<g_e\left(W^{n+1},W_e^{n+1}\right)-g_e\left(W^n,W_e^n\right),v_e{>}_{\mathrm{\Upsilon }}-\left(d_t{\eta }_e^{n+1},v_e\right) \\ +\left(d_t{u}_e^{n+1}-{\partial }_t{u}_e^{n+1},v_e\right)+\lambda \left({\eta }_e^{n+1},v_e\right). \end{gathered}$$

Subtracting (63) from (64), letting ξⁿ⁺¹ = Wⁿ⁺¹−Uⁿ⁺¹, ${\xi }_b^{n+1}=W_b^{n+1}-U_b^{n+1}$, ${\xi }_e^{n+1}=W_e^{n+1}-U_e^{n+1}$, and choosing $v={\xi }^{n+1},v_b={\xi }_b^{n+1}$, $v_e={\xi }_e^{n+1}$ with the definition $W_h(t,x):=\sum _{i=0}^n{\varphi }_i(t)W\left(t_i,x\right)$ and Lemma 2.2, we have the following equations (65), (66) and (67) respectively

$$\begin{gathered} \frac{{\Vert {\xi }^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }^n\Vert }^2}{2\mathrm{\Delta }t}+D_c{\Vert \nabla {\xi }^{n+1}\Vert }^2 \\ \le C_1\left({\Vert {\xi }^n\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert {\xi }^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^{t_n}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_1\left({\Vert {\xi }_b^n\Vert }^2+{\Vert {\xi }^n\Vert }^2+{\Vert {\xi }^{n+1}\Vert }^2\right) \\ +C_1\left({\Vert {\xi }^{n+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert {\xi }^{n+1}\Vert }_{L^2({\rm Y})}^2\right)+E^{n+1}\left({\xi }^{n+1}\right) \\ +C_1{\int }_0^{t_n}{\Vert \left(W-W_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s, \end{gathered}$$ (65)

$$\begin{gathered} \frac{{\Vert {\xi }_b^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }_b^n\Vert }^2}{2\mathrm{\Delta }t}+D_b{\Vert \nabla {\xi }_b^{n+1}\Vert }^2 \\ \le C_1\left({\Vert {\xi }_b^n\Vert }^2+{\Vert {\xi }^n\Vert }^2+{\Vert {\xi }_b^{n+1}\Vert }^2\right)+E_b^{n+1}\left({\xi }_b^{n+1}\right), \end{gathered}$$ (66)

$$\begin{gathered} \frac{{\Vert {\xi }_e^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }_e^n\Vert }^2}{2\mathrm{\Delta }t}+D_e{\Vert \nabla {\xi }_e^{n+1}\Vert }^2 \\ \le C_3\left({\Vert {\xi }^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e^n\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^{t_n}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_3{\Vert {\xi }_e^{n+1}\Vert }^2+E_e^{n+1}\left({\xi }_e^{n+1}\right) \\ +C_3{\int }_0^{t_n}{\Vert \left(W-W_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s. \end{gathered}$$ (67)

Adding the equations (65), (66) and (67) as n varies leads to equations (68), (69) and (70) correspondingly

$$\begin{gathered} \frac{{\Vert {\xi }^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }^0\Vert }^2}{2\mathrm{\Delta }t}+D_c\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2 \\ \le C_1\sum _{l=0}^n\left({\Vert {\xi }^l\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert {\xi }^l\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e^l\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^{t_l}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_1\sum _{l=0}^n\left({\Vert {\xi }_b^l\Vert }^2+{\Vert {\xi }^l\Vert }^2+{\Vert {\xi }^{l+1}\Vert }^2\right) \\ +C_1\sum _{l=0}^n\left({\Vert {\xi }^{l+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert {\xi }^{l+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right)+\sum _{l=0}^n{E}^{l+1}\left({\xi }^{l+1}\right) \\ +C_1\sum _{l=0}^n{\int }_0^{t_l}{\Vert \left(W-W_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s, \end{gathered}$$ (68)

$$\begin{gathered} \frac{{\Vert {\xi }_b^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }_b^0\Vert }^2}{2\mathrm{\Delta }t}+D_b\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_b^l\Vert }^2 \\ \le C_1\sum _{l=0}^n\left({\Vert {\xi }_b^l\Vert }^2+{\Vert {\xi }^l\Vert }^2+{\Vert {\xi }_b^{l+1}\Vert }^2\right)+\sum _{l=0}^n{E}_b^{l+1}\left({\xi }_b^{l+1}\right), \end{gathered}$$ (69)

$$\begin{gathered} \frac{{\Vert {\xi }_e^{n+1}\Vert }^2}{2\mathrm{\Delta }t}-\frac{{\Vert {\xi }_e^0\Vert }^2}{2\mathrm{\Delta }t}+D_e\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_e^l\Vert }^2 \\ \le C_3\sum _{l=0}^n\left({\Vert {\xi }^l\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\Vert {\xi }_e^l\Vert }_{L^2(\mathrm{\Upsilon })}^2+{\int }_0^{t_l}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\right) \\ +C_3\sum _{l=0}^n{\Vert {\xi }_e^{l+1}\Vert }^2+\sum _{l=0}^n{E}_e^{l+1}\left({\xi }_e^{l+1}\right)+C_3\sum _{l=0}^n{\int }_0^{t_l}{\Vert \left(W-W_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s. \end{gathered}$$ (70)

In the right hand side of (68), the following estimates (71) to (74) can be obtained. Notice that each ϕ_i(s), i = 0, ⋯, l, has a compact support and the product ϕ_i(s)ϕ_j(s) = 0, if |j − i| > 1, so that we have

$$\begin{gathered} {\int }_0^{t_l}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s={\int }_0^{t_l}{\Vert \sum _{i=0}^l{\varphi }_i(s){\xi }^i\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s \\ \le {\int }_0^{t_l}3\sum _{i=0}^l{\left|{\varphi }_i(s)\right|}^2{\Vert {\xi }^i\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s \\ \le 2\mathrm{\Delta }t\sum _{i=0}^l{\Vert {\xi }^i\Vert }_{L^2(\mathrm{\Upsilon })}^2, \end{gathered}$$ (71)

where ${\int }_0^{t_l}3{\left|{\varphi }_i(s)\right|}^2\text{d}s\le 2\mathrm{\Delta }t$. Then from (71), with the Trace Theorem, we get

$$\begin{gathered} \sum _{l=0}^n{\int }_0^{t_l}{\Vert \left(W_h-U_h\right)(s)\Vert }_{L^2(\mathrm{\Upsilon })}^2\text{d}s\le \sum _{l=0}^{n}2\mathrm{\Delta }t\sum _{i=0}^n{\Vert {\xi }^i\Vert }_{L^2(\mathrm{\Upsilon })}^2 \\ \le 2T\sum _{i=0}^n{\Vert {\xi }^i\Vert }_{L^2(\mathrm{\Upsilon })}^2 \\ \le C\sum _{l=0}^n\left({\Vert {\xi }^l\Vert }^2+{\Vert \nabla {\xi }^l\Vert }^2\right). \end{gathered}$$ (72)

By Lemma 2.3, we have

$$\begin{gathered} \mathrm{\Delta }t{C}_1\sum _{l=0}^n\left({\Vert {\xi }^{l+1}\Vert }_{L^2(\partial \mathrm{\Omega })}^2+{\Vert {\xi }^{l+1}\Vert }_{L^2(\mathrm{\Upsilon })}^2\right)\le \frac{\mathrm{\Delta }t{D}_c}{8}\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2 \\ +\mathrm{\Delta }tC{\Vert {\xi }^{n+1}\Vert }^2+\mathrm{\Delta }tC\sum _{l=1}^n{\Vert {\xi }^l\Vert }^2, \end{gathered}$$ (73)

the first and second terms in the right hand side of (73) can be canceled in (68) if Δt is sufficiently small, however, this small Δt doesn’t depend on the spatial mesh size. Then with Lemma 2.2, Lemma 2.3, by the estimates of ∂_tη, ∂_tη_e in Theorem 4.1 and the estimates for ∂_tη_b, which is easier to obtain from the definition of W_b in Section 5, we know ∂_tW, ∂_tW_e and ∂_tW_b are bounded with H¹ norm, so that

$$\begin{gathered} \sum _{l=0}^n{E}^{l+1}\left({\xi }^{l+1}\right)\le C\mathrm{\Delta }t+C\frac{h^{2k}}{\mathrm{\Delta }t}+\frac{D_c}{8}\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2 \\ +C{\Vert {\xi }^{n+1}\Vert }^2+C\sum _{l=1}^n{\Vert {\xi }^l\Vert }^2, \end{gathered}$$ (74)

where C is a positive constant and doesn’t depend on n. Other terms on the right-hand side of (68) can be treated similarly.

So that from (68) and the estimates (72) to (74), we have (75)

$$\begin{gathered} \frac{1}{2}{\Vert {\xi }^{n+1}\Vert }^2+\frac{3D_c}{4}\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2 \\ \le C_{c1}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }^l\Vert }_{L^2(\mathrm{\Omega })}^2+C_{c2}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }_b^l\Vert }^2+C_{c3}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }_e^l\Vert }^2 \\ +\frac{D_e}{8}\mathrm{\Delta }t\sum _{l=0}^n{\Vert \nabla {\xi }_b^l\Vert }^2+C_{c4}{\Vert {\xi }^0\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+C_c\left(\mathrm{\Delta }{t}^2+h^{2k}\right). \end{gathered}$$ (75)

From (69), with similar estimates as (68), we have (76)

$$\begin{gathered} \frac{1}{2}{\Vert {\xi }_b^{n+1}\Vert }^2+D_b\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_b^l\Vert }^2 \\ \le C_{b1}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }_b^l\Vert }^2+C_{b2}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }^l\Vert }^2+C_{b3}{\Vert {\xi }_b^0\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+C_b\left(\mathrm{\Delta }{t}^2+h^{2k}\right). \end{gathered}$$ (76)

From (70), with similar estimates as (68), we have (77)

$$\begin{gathered} \frac{1}{2}{\Vert {\xi }_e^{n+1}\Vert }^2+\frac{3D_e}{4}\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_e^l\Vert }^2 \\ \le C_{e1}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }^l\Vert }^2+C_{e2}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }_e^l\Vert }^2+\frac{D_c}{8}\mathrm{\Delta }t\sum _{l=0}^n{\Vert {\xi }^l\Vert }^2 \\ +C_{e3}{\Vert {\xi }_e^0\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2+C_e\left(\mathrm{\Delta }{t}^2+h^{2k}\right). \end{gathered}$$ (77)

C_c, C_b, C_e in (75),(76) and (77) do not depend on n.

Summing (75), (76) and (77), we have

$$\begin{gathered} {\Vert {\xi }^{n+1}\Vert }^2+{\Vert {\xi }_b^{n+1}\Vert }^2+{\Vert {\xi }_e^{n+1}\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_b^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_e^l\Vert }^2 \\ \le C_1\mathrm{\Delta }t\sum _{l=0}^n\left({\Vert {\xi }^l\Vert }^2+{\Vert {\xi }_b^l\Vert }^2+{\Vert {\xi }_e^l\Vert }^2\right) \\ +C_2\left({\Vert {\xi }^0\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+{\Vert {\xi }_b^0\Vert }_{H^1\left({\mathrm{\Omega }}_c\right)}^2+{\Vert {\xi }_e^0\Vert }_{H^1\left({\mathrm{\Omega }}_e\right)}^2+\mathrm{\Delta }{t}^2+h^{2k}\right). \end{gathered}$$

If Δt is small enough, by discrete Gronwall’s Inequality, the estimate below follows

$$\begin{gathered} {\Vert {\xi }^{n+1}\Vert }^2+{\Vert {\xi }_b^{n+1}\Vert }^2+{\Vert {\xi }_e^{n+1}\Vert }^2 \\ +\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_b^l\Vert }^2+\mathrm{\Delta }t\sum _{l=1}^{n+1}{\Vert \nabla {\xi }_e^l\Vert }^2\le C\left(\mathrm{\Delta }{t}^2+h^{2k}\right), \end{gathered}$$

where C does not depend on n, Δt and h. Then by eⁿ = ξⁿ + ηⁿ, $e_b^n={\xi }_b^n+{\eta }_b^n$, $e_e^n={\xi }_e^n+{\eta }_e^n$, and Theorem 4.1, we complete the proof. □

7. Numerical Experiments

In this section, we illustrate the convergence theorem for the fully discrete scheme (63) using Examples 1 and 2 below, and then apply the methodology to show the existence of Ca²⁺ wave propagation numerically in Example 3 and 4. The ODE system (12) plays the key role for calcium wave initiation and propagation, which is solved by backward Euler’s method. The numerical schemes in this section are implemented in FreeFem++, see [31]. All presented examples are in 2D, but theorems and simulations are also valid in 3D.

Example 1.

In this problem we cosider two coupled PDEs with the unknowns u and u_e:

$$\{\begin{array}{l}{\partial }_{t}u-\mathrm{\Delta }u=f_1(x,y,t)\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_t{u}_e-\mathrm{\Delta }{u}_e=f_2(x,y,t)\text{on}{\mathrm{\Omega }}_e\times (0,T]\hfill \end{array}$$ (78)

where T = 1.3 and the boundary conditions are: ∂_nu = g₁(x, y, t) on ∂Ω×(0, T] and

$${\partial }_{n}u=P(t,u)\left(u_e-u\right)+g_2(x,y,t)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (79)

$${\partial }_n{u}_e=P(t,u)\left(u-u_e\right)+g_3(x,y,t)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (80)

Here, let (n_x, n_y) be the unit outer normal vector on ∂Ω_c for g₁, g₂; on Υ for g₃, then the exact solution u, u_e and corresponding functions are listed below:

u	$e^{\frac{x^2+y^2+4t}{4}}/10$
u e	ex+y(sint + 2)/8
f 1	$-e^{x^2/4+y^2/4+t}\left(x^2+y^2\right)/40$
f 2	ex+y(cos(t) − 2(sin(t) + 2))/8
g 1	$n_x{e}^{\frac{x^2+y^2+4t}{4}}x/20+n_y{e}^{\frac{x^2+y^2+4t}{4}}y/20$
g 2	$n_x{e}^{\frac{x^2+y^2+4t}{4}}x/20+n_y{e}^{\frac{x^2+y^2+4t}{4}}y/20-P(t,u)\left(u_e-u\right)$
g 3	nxex+y(sint + 2)/8 + nyex+y(sint + 2)/8 + P(t,u)(ue − u)

The coefficients in ODE (12) are taken from [17]:

$$k_a^{-}=28.8,k_a^{+}=1500,k_b^{-}=385.9,k_b^{+}=1500,k_c^{-}=0.1,k_c^{+}=1.75$$ (81)

The initial conditions for (12) are chosen as: c₁(0) = 0.5, o(0) = 0, c₂(0) = 0.5.

Example 2.

In this example we consider three coupled PDEs with unknowns u, b and u_e:

$$\{\begin{array}{l}{\partial }_{t}u-\mathrm{\Delta }u=f_1(x,y,t)-bu\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_{t}b-\mathrm{\Delta }b=f_2(x,y,t)-bu\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_t{u}_e-\mathrm{\Delta }{u}_e=f_3(x,y,t)\text{on}{\mathrm{\Omega }}_e\times (0,T]\hfill \end{array}$$ (82)

where T = 1.3, and the boundary conditions are: ∂_nu = g₁(x, y, t) on ∂Ω×(0, T], ∂_nb = g₂(x, y, t) on ∂Ω_c × (0, T] and

$${\partial }_{n}u=P(t,u)\left(u_e-u\right)-\frac{u}{(1+u)u_e}+g_3(x,y,t)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (83)

$${\partial }_n{u}_e=P(t,u)\left(u-u_e\right)+\frac{u}{(1+u)u_e}+g_4(x,y,t)\text{on}{\rm T}\times (0,T]$$ (84)

Here, let (n_x, n_y) be the unit outer normal vector on ∂Ω_c for g₁, g₂, g₃; on Υ for g₄, then the exact solution u, b, u_e and corresponding functions are listed below:

b	e xyt/16
u	$e^{\frac{x^2+y^2+4t}{4}}/10$
u e	ex+y(sint + 2)/16 + 1
f 1	$-e^{x^2/4+y^2/4+t}\left(x^2+y^2\right)/40+bu$
f 2	−exyt/16(−xy + t2(x2 + y2)/16)/16 + bu
f 3	ex+y(cos(t) − 2(sin(t) + 2))/8
g 1	$n_x{e}^{\frac{x^2+y^2+4t}{4}}x/20+n_y{e}^{\frac{x^2+y^2+4t}{4}}y/20$
g 2	$n_x{e}^{xyt/16}ty/16+n_y{e}^{xyt/16}tx/16$
g 3	$n_x{e}^{\frac{x^2+y^2+4t}{4}}x/20+n_y{e}^{\frac{x^2+y^2+4t}{4}}y/20-P(t,u)\left(u_e-u\right)+\frac{u}{(1+u)u_e}$
g 4	$n_x{e}^{x+y}(\sin t+2)/8+n_y{e}^{x+y}(\sin t+2)/8+P(t,u)\left(u_e-u\right)-\frac{u}{(1+u)u_e}$

The ODE system and its initial conditions are the same as in Example 1. Example 1 and 2 share the same space-time meshes (see Figure 2 for examples of spatial meshes). The spatial mesh sizes are: π/8, π/16, π/32, π/64, π/128. Letting Δt be the time step size and h the spatial mesh size, we have the relation between the two defined as Δt = Ch², where C = 32T/(5π²), T = 1.3. Both examples have exact solutions. We define the error in the L² norm for u as: $\Vert u-u_h\Vert ={\sum }_{i=1}^N\Vert u\left(t_i\right)-u_h\left(t_i\right)\Vert /N$ and the error in H¹ semi-norm as: $\Vert \nabla u-\nabla u_h\Vert ={\sum }_{i=1}^N\Vert \nabla u\left(t_i\right)-\nabla u_h\left(t_i\right)\Vert /N$ where N = T/Δt, t_i = iΔt, u is the exact solution, u_h is the numerical solution. Then let b_h, $u_e^h$ be the numerical solutions, the errors are defined similarly. In Figure 7, we show the convergence rates for P₁ elements in space, which are optimal.

Figure 2:

Meshes in which the blue region is Ω_c, the brown region is Ω_e: (a) coarse with mesh size π/8; (b) refined mesh with mesh size π/16. The radius of the larger circle is 2, radius of the smaller one is 1.

Figure 7:

The blue region as in (l) is ${\overline{\mathrm{\Omega }}}_c$, the value is always 0 on ${\overline{\mathrm{\Omega }}}_c/\mathrm{\Upsilon }$. Red color part on the inner circle (ER membrane) means the RyR channels are open. We can see that from (a)-(g), the open state propagates to the whole ER membrane which is faster compared with Example 3. From (h) to (l), the value of open probability decreases to the close state.

Example 3.

In this example we present a minimal system that produces Ca²⁺ waves. Units were adjusted so that t has unit s, u, u_e have unit μM:

$$\{\begin{array}{l}{\partial }_{t}u-\mathrm{\Delta }u=0\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_t{u}_e-\mathrm{\Delta }{u}_e=0\text{on}{\mathrm{\Omega }}_e\times (0,T]\hfill \end{array}$$ (85)

where T = 12 and the boundary conditions are:

$${\partial }_{n}u=C_3(1000-u)-C_2\frac{u}{1+u}-C_1\frac{u^2}{1+u^2}+f(x,y,t)\text{on}\partial \mathrm{\Omega }\times (0,T]$$ (86)

$${\partial }_{n}u=C_1^{e}P(t,u)\left(u_e-u\right)-C_2^e\frac{u}{(2+u)u_e}+C_3^e\left(u_e-u\right)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (87)

$${\partial }_n{u}_e=C_1^{e}P(t,u)\left(u-u_e\right)+C_2^e\frac{u}{(2+u)u_e}-C_3^e\left(u_e-u\right)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (88)

The initial conditions are u(x, y, 0) = 0.05, u_e(x, y, 0) = 180, the ODE system is the same as in Example 1, and the initial conditions of the ODE are c₁(0) = 0.798, o(0) = 0, c₂(0) = 0.202. In (86), the value 1,000 is the extracellular Ca²⁺ concentration, and f is a calcium influx function: f(x, y, t) = 3 if 0.05 ≤ t ≤ 0.65 and y − x ≥ 2.5; f(x, y, t) = 0 elsewhere. The coefficients in (86) to (88) are: $C_1^e=0.17$, $C_3^e=1/150$, $C_2^e=8853.54$, C₃ = 1/540000, C₁ = C₂ = 19954C₃.

Example 3 is constructed to show the initiation and propagation of a calcium wave which plays a critical role in neuronal signal processing, see Figure 4. For the computation, we use a similar geometry as in Examples 1 and 2. The radii of the two circles, with center (0,0), are 1 and 2, but with different meshes and elements, where T = 12, Δt = 0.00375, and the spatial mesh size is h = π/24 with P₃ elements in space. With the help of Figure 4, a calcium wave can be described as follows. An extracellular stimulus produces Ca²⁺ influx across the outer interface (the plasma membrane) raising the calcium concentration in parts of cytosol (Ω_c) and ER (Ω_e) see Figures 4(a), 4(b). In this example, the influx goes from 0.05s to 0.65s. Then, an increased concentration activates the release of Ca²⁺ from the ER at t = 0.72s, see Figure 4(c), which in turn generates the calcium spike and thus mediates global activation of the cell, see Figures 4(d) to 4(g). The calcium concentration reaches its peak around 3.12s in Figure 4(h), from the color bar 4(m), u varies from 0.05 μM to value greater than 1.5 μM. Meanwhile, u_e decreases to the value around 176 μM. After reaching the peak, u decreases and u_e increases to the initial state, see 4(i) to 4(l). The term (10) is essential for generating calcium wave, without the ODE system, there is no calcium wave. Figure 5 shows the open probability function P(t, u) in equation (11) for RyR channels on the ER membrane. It ranges from 0 to 0.81. Instability of the scheme (63) can be observed with time step size larger than 0.00375.

Figure 4:

Initiation (a)-(b), propagation (c)-(h) and recovery (i)-(l) of the calcium wave in a 2D cell. As in (l) - an equilibrium state, the blue region is cytosol (Ω_c), the brown region is ER (Ω_e). u and u_e are the calcium concentrations in cytosol and ER respectively.

Figure 5:

The blue region as in (l) is ${\overline{\mathrm{\Omega }}}_c$, the value is always 0 on ${\overline{\mathrm{\Omega }}}_c/\mathrm{\Upsilon }$. Red color part on the inner circle (ER membrane) means the RyR channels are open. We can see that from (a)-(h), the open state propagates to the whole ER membrane which leads to the release of calcium from ER. From (i) to (l), the value of open probability decreases to the equilibrium state.

Example 4.

In this example we present a full system with different coefficients, taken from [2], which produces Ca²⁺ waves. Units were adjusted so that t has unit s, u, u_e have unit μM:

$$\{\begin{array}{l}{\partial }_{t}u-220\mathrm{\Delta }u=f(b,u)\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_{t}b-20\mathrm{\Delta }b=f(b,u)\text{on}{\mathrm{\Omega }}_c\times (0,T]\hfill \\ {\partial }_t{u}_e-220\mathrm{\Delta }{u}_e=0\text{on}{\mathrm{\Omega }}_e\times (0,T]\hfill \end{array}$$ (89)

where T = 80, $f(b,u)=K_b^{-}\left(b^0-b\right)-K_b^{+}bu$, ∂_nb = 0 on ∂Ω_c, and other boundary conditions are:

$${\partial }_{n}u=C_3(1000-u)-\frac{C_{2}u}{1.8+u}-\frac{C_1{u}^2}{{0.06}^2+u^2}+g(x,y,t)\text{on}\partial \mathrm{\Omega }\times (0,T]$$ (90)

$${\partial }_{n}u=C_1^{e}P(t,u)\left(u_e-u\right)-C_2^e\frac{u}{(0.18+u)u_e}+C_3^e\left(u_e-u\right)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (91)

$${\partial }_n{u}_e=C_1^{e}P(t,u)\left(u-u_e\right)+C_2^e\frac{u}{(0.18+u)u_e}-C_3^e\left(u_e-u\right)\text{on}\mathrm{\Upsilon }\times (0,T]$$ (92)

The initial conditions are u(x, y, 0) = 0.05, b(x, y, 0) = 37, u_e(x, y, 0) = 250, in f(b, u), b⁰ = 40, $K_b^{-}=16.65$, $K_b^{+}=27$. The ODE system is the same as in Example 1, and the initial conditions of the ODE are c₁(0) = 0.994, o(0) = 1.5721 × 10⁻⁷, c₂(0) = 5.6625 × 10⁻³. In (90), the value 1,000 is the extracellular Ca²⁺ concentration, and g is a calcium influx function: $g\text{(}x\text{,}y\text{,}t\text{)=}240e^{-0.01/\left(0.01-{(t-0.2)}^2\right)+1}$ if 0.1 < t < 0.3 and y − x ≥ 2.5; g(x, y, t) = 0 elsewhere. The coefficients in (90) to (92) are: $C_1^e=0.829468$, $C_2^e=11000$, $C_3^e=0.038$, C₁ = 8.5, C₂ = 37.6, C₃ = 0.0045.

Example 4 is constructed to show the initiation and propagation of the calcium wave in a full model, see Figure 6. For computation, we use a similar geometry as in Examples 3. The radii of the two circles, with center (0,0), are 1.2 and 2, but with different meshes and P₁ elements, where T = 80, Δt = 0.01/16, and the spatial mesh size is h = π/32. Due to larger diffusion coefficients and the buffer b, propagation of the calcium is much faster than Example 3, but the recovery is slower. Here, we don’t show the graph of b which varies from 2 to 38, since it’s less important. Figure 7 shows the open probability function P(t, u) in equation (11) for RyR channels on the ER membrane. It ranges from 0 to 0.96. Instability of the scheme (63) can be observed with time step size larger than 0.01/16.

Figure 6:

Initiation (a)-(b), propagation (c)-(e) and recovery (f)-(p) of the calcium wave in a 2D cell. As in (a), (p) - the equilibrium state, the blue region is cytosol (Ω_c), the dark-red region is ER (Ω_e). u and u_e are the calcium concentrations in cytosol and ER respectively.

8. Conclusion

In this paper, we analyze the model of calcium dynamics in neurons with ER, obtain the existence, uniqueness and boundedness of the solution, and then propose an efficient implicit-explicit finite element scheme. The necessity of the ODE systems on interfaces is shown in Section 7. The focus on calcium dynamics is motivated by the fact that intracellular calcium signals in response to electrical events (e.g. action potentials) trigger a multitude of calcium regulated processes which are relevant in cellular development, learning, and cell survival. The complexity of the cellular calcium-regulating machinery typically prohibits a systematic experimental study and computational models are highly relevant in studying the effect of morphological and biophysical changes on calcium dynamics. Models, theorems and algorithms are well established for electrical models, however, calcium dynamics has not been extensively studied, partly because lower-dimensional approximations are insufficient and detailed, high-resolution PDE-based simulations are required (integration of complex geometric structures of cells and intracellular organelles). Thus, 3D models are necessary to accurately capture calcium dynamics in cells and high-performance computing must be utilized. The L² error estimates, high order stable multistep implicit-explicit schemes and a parallel implementation of the numerical methods for the 3D problem are part of ongoing work.

Figure 3:

Both examples employ piecewise continuous P₁ element in space, for all solutions the errors with L² norm have rate 2, the errors with H¹ semi-norm have rate 1.

Highlights.

• The existence, uniqueness and boundedness of the solution are proved for the model of calcium dynamics in neurons, which is governed by coupled diffusion-reaction systems with highly nonlinear ODE-controlled interfaces.

• An efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms is proposed and analyzed. The stability does not depend on the spatial mesh size and the optimal convergence rate in H¹ norm is obtained.

• Simulations are given to show the initiation and propagation of a calcium wave which plays a critical role in neuronal signal processing. The ODE-controlled flux is essential for generating calcium wave, without the ODE system, there is no calcium wave.