Domain decomposition and mortar mixed approach for nonlinear elliptic equations modeling flow in porous media

Abstract

The equations describe the behavior of steady state flow in porous medium generally results in elliptic partial differential equations with coefficient represents the permeability of the medium. This article presents the extension of mortar mixed method for second order nonlinear elliptic equations that describes flow in porous media. The domain is decomposed into non-overlapping regions with each partitioned independently. The grids on subdomains are allowed to be non-matching across the subdomains internal boundaries. The fixed point argument (FPA) is employed to establish the existence and uniqueness of discrete problem, and optimal order error estimates are provided for approximations. The computational results are given to validate the theory.

1. Introduction

The nonlinear partial differential equations have become of fundamental importance in computational science and engineering. Besides other important applications, the nonlinear partial differential equations appear in the field of dynamics, finance and quantum mechanics. In modern scientific approach, many complicated industrial problems require accurate and efficient numerical methods to support and guide the theoretically established results. The remarkable development in technology provided sufficient resources to develop fast numerical methods with precise algorithms which are capable to deal with the solution of challenging models in many practical applications.

Let Ω be a polygonal domain in ${\mathit{R}}^2$ with boundary ∂Ω, we consider second order nonlinear elliptic equation in divergence form

$$-\mathrm{\nabla }\cdot (\mathcal{A}(\mathit{x},p(\mathit{x}))\mathrm{\nabla }p(\mathit{x})=f(\mathit{x},p(\mathit{x}))\mathrm{\text{in}}\mathrm{\Omega },$$ (1.1)

with boundary condition

$$p(\mathit{x})=g(\mathit{x}),\mathrm{\text{on}}\partial \mathrm{\Omega },$$ (1.2)

where $\mathit{x}=(x,y)$. Introducing the flux variable $\mathit{u}=-\mathcal{A}(p)\mathrm{\nabla }p$, the mixed form of (1.1) – (1.2) is given by

$$\mathit{u}=-\mathcal{A}(\mathit{x},p(\mathit{x}))\mathrm{\nabla }p,\mathrm{\text{in}}\mathrm{\Omega },$$ (1.3)

$$\mathrm{\nabla }\cdot \mathit{u}=f(\mathit{x},p(\mathit{x})),\mathrm{\text{in}}\mathrm{\Omega },$$ (1.4)

$$p=g,\mathrm{\text{on}}\partial \mathrm{\Omega }.$$ (1.5)

For the sake of simplicity, we shall drop the variable x in the notation below. Equations (1.3) – (1.4) represents Darcy velocity and mass conservation respectively. The coefficient $\mathcal{A}(p)$ denotes pressure dependent permeability. We assume that $\mathcal{A}(p)$ has continuous derivatives up order two such that

$$0<a\le \mathcal{A}(p)\le b<\infty .$$

For the purpose of simplicity, we examine the Dirichlet boundary condition. However a more general boundary condition can be applied in a similar way. The formulation framework can be adjusted to apply other type of boundary condition. We assume that the problem has $H^{2+ϵ},0<ϵ<1$ regularity where $H^r$ denote the standard Sobolev space. We have $H^{2+ϵ}$ regularity, for example, $f\in H^{ϵ}(\mathrm{\Omega })$, $g\in H^{3/2+ϵ}(\partial \mathrm{\Omega })$ and domain Ω is smooth enough [30], [27], [38].

Darcy law has many important applications including water flow through an aquifer. Also the Darcy law combined with equation of mass conservation models ground water flow which play a crucial role in hydrology. Applications in hydrology include the disposal of nuclear waste, ground water transport and soil drainage etc. In many applications, it is important to have careful risk assessment, for example, nuclear waste disposal. Thus the accurate and efficient approximation is required. Another crucial application of subsurface flow is petroleum engineering. In petroleum reservoir, Darcy law interprets oil water and gas flows. Other than geomechanical engineering, porous media flows appeared in many industrial applications, e.g., biomedical modeling, filtration technology, etc.

The numerical approximation of the mathematical models describing subsurface flow is highly effected by the physical properties of the medium, for example, permeability and porosity in porous media flow [10], [37], [45]. Permeability measures the ability of medium to transmit fluid through a specific point of domain which fluctuate over small distances with the variation in properties of the surface. The permeability also varies when the region of interest consists of rocks. The traditional approaches to tackle such problems require a very fine mesh to resolve the variation in physical attributes for an accurate approximation which yield a huge system of algebraic equations [42], [32]. The main difficulty in using direct approaches is the computational cost. Therefore, it is necessary to adopt some alternate approaches which provide accuracy and efficiency by reducing the computational complexity of the system of equations encountered by the corresponding technique. In recent years, the efficient and accurate approximation of the mathematical models in geomechanics have been grown to large extent. Particularly, the numerical solution of the equations modeling fluid flow in porous media has become of interest due to their applications in diverse fields.

The domain decomposition methods referred as divide and conquer techniques have been widely used to handle these difficulties. The domain decomposition methods reduce the computational burden by dividing the domain and solving the partial differential equation over small regions. The main advantages of domain decomposition methods are their capability to handle the problems posed on complex physical domains, ability to deal with the partial differential equations possessing different behavior on different regions of the domain, and the enhancement of parallel computation. Several types of domain decomposition methods have been proposed and analyzed [1], [21], [22], [11], [29].

The mixed finite element methods are proven to be highly effective in flow related problems due to their physical attributes such as mass conservation property. In the approximation of physical models related to fluid dynamics, the mixed finite element methods provide the accurate representation of conservation. Moreover, the mixed methods are capable to approximate two unknown simultaneously with same accuracy. The mixed finite element methods have been widely considered for many areas, especially the flow problems [3], [6], [8], [36], [9], [42]. Moreover, the mixed methods have been extensively analyzed and implemented for elliptic [49], [43], [15], [13], [50], [16], [40], [41], [44], [35], [14], [20], [23], [34] and parabolic [4], [17], [28], [33] partial differential equations.

In 2000, Arbogast et al. [5] invented the multiblock methodology based on domain decomposition and mortar method. In this technique the domain is divided into several small regions separated by interfaces. The subdomains and interface are partitioned on same scale and mixed finite element method is applied on each block to solve the subproblems locally. Comparing to the other domain decomposition methods, the mortar mixed method offers great flexibility to use different models on the different subdomain blocks [26], [48] and different numerics [47], [48] and it allows to use the adaptive techniques [52] to improve the accuracy locally where needed. Moreover, this approach is suitable to handle complex geometry and discontinuous coefficients. Furthermore, the linear system encountered by this formulation can be easily implemented in parallel by using parallel domain decomposition algorithm which enhance the efficiency of method. The mixed method methods on locally refined grids [24], [25] employ the concept of slave and worker nodes to impose continuity of flux on internal boundaries and allow nested grids but the grids on neighboring subdomains are required to matching on interface. The mortar mixed technique provides better approximation on non-matching grids. Since the adjacent regions on domain are discretized using different grids, therefore, the normal trace of velocity space is not available as Langrange multiplier space. Hence a finite element space named mortar space is constructed on interface to couple the global problem. This method gives optimal order convergence provided that the interface space has one order higher approximation than the normal trace of velocity space. This method has been further applied to different equations [31], [51], [7], [9], [53].

This study is dedicated to the extension of mortar mixed method for second order elliptic equations with pressure dependent coefficient. We provide the analysis and the implementation of the method for the model problem under consideration. Observe that the pressure dependence of coefficient turns (1.3) into nonlinear Darcy equation. Thus to establish the existence of solution for discrete problem, we follow the fixed-point argument developed in [40], [44]. To apply this argument, we construct a map from an appropriate function space to itself, and thus the existence of a solution for the discrete problem follows from the Brouwer fixed-point theorem. Next, we derive the optimal order convergence for both velocity and pressure approximations. We propose a pressure dependent interface formulation and provide an error bound for mortar interface pressure. We provided the implementation procedure for our method and presented the algorithms for iterative methods. Fixed-point iteration and Newton Raphson methods are used in implementation to solve discretized nonlinear system. We implemented the method on four, eight and sixteen subdomains and presented the numerical results.

For sake of convenience and clarity of presentation, we now introduce the following notations and symbols. For a domain $D\subset {\mathit{R}}^2$, the $L^2(D)$ and ${(L^2(D))}^d$ inner product is denoted by ${(\cdot ,\cdot )}_D$. Similarly the $L^2(D)$ or ${(L^2(D))}^2$ norm is denoted by ${\Vert \cdot \Vert }_D$. For a non-negative integer m or $1\le q<\infty$, $W^{m,q}(D)$ represent the Sobolev space. The norm and semi norm on Sobolev space is denoted by ${\Vert \cdot \Vert }_{m,q,D}$ and $|\cdot {|}_{m,q,D}$ respectively. The notation ${\langle \cdot ,\cdot \rangle }_B$ and ${\Vert \cdot \Vert }_B$ represent the $L^2$ inner product and norm on the boundary B of domain D. Moreover, $\mathit{H}(div,D)$ denote the square integrable functions whose weak divergence is also $L^2(D)$, that is, $\mathit{H}(div,D)=\{\mathit{u}\in L^2(D):\mathrm{\nabla }\cdot \mathit{u}\in L^2(D)\}$. Hereafter, subscript in the above notation will be dropped if $D=\mathrm{\Omega }$. Finally, the vectors will be denoted by boldface letters.

The remaining of the study is organized as follows. The mathematical foundation needed to create the method in later sections is presented in next section. We give the projection operators and weak velocities in Section 3. The discrete form of the method is presented in Section 4. Section 5 consist of The unique solvability of discrete system. The optimal order convergence for approximations is derived in Section 6. Section 7 is dedicated to proof of uniqueness of solution for the discrete system. Section 8 provides interface formulation and error bound for interface pressure approximation. The implementation procedure along with numerical results is presented in Section 9. Finally, the findings of article are concluded in Section 10.

2. Mathematical preliminaries

Our method is based on domain decomposition so we begin by decomposing the domain into finite number of small blocks which restrict the computational cost. Let us divide the domain into nb subdomain blocks $\{K_i,1\le i\le nb\}$ such that $\overline{\mathrm{\Omega }}={\cup }_{i=1}^{nb}{K}_i$ which are non-overlapping, that is, $K_i\cap K_j=\varnothing ,i\ne j$. Let

$${\gamma }_{ij}=\partial K_i\cap \partial K_j,\gamma ={\cup }_{1\le i<j\le nb}{\gamma }_{ij},{\gamma }_i=\partial K_i\cap \gamma =\partial K_i\setminus \partial \mathrm{\Omega },$$

denote the interface between two subdomains, the whole interface and the interface related to the $i^{th}$ subdomain. In sense of domain decomposition, we define the weak spaces as follows

$${\mathit{U}}_i=\mathit{H}(div;K_i),S_i=L^2(K_i),{\mathcal{L}}_{ij}=H^{1/2}({\gamma }_{ij}),\mathit{U}={\oplus }_{i=1}^{nb}{\mathit{U}}_i,S={\oplus }_{i=1}^{nb}{S}_i=L^2(\mathrm{\Omega }),\mathcal{L}={\oplus }_{1\le i<j\le nb}{\mathcal{L}}_i.$$

We define the norm on U as follows

$${\Vert \mathit{v}\Vert }_{\mathit{U}}={({\Vert \mathit{v}\Vert }^2+{\Vert \mathrm{\nabla }\cdot \mathit{v}\Vert }_{\tilde{\mathrm{\Omega }}}^2)}^{1/2}\mathrm{\text{where}}{\Vert \cdot \Vert }_{k,\tilde{\mathrm{\Omega }}}={\left(\sum _{i=1}^{nb}\right({\Vert \cdot \Vert }_{k,K_i}^2\left)\right)}^{1/2}.$$

We partition each $K_i$ independently into a conforming quasi-uniform triangulation ${\mathcal{T}}_h^i$ such that the maximal element diameter is $h_i$, for $1\le i\le nb$. For the analysis, we set $h={\max }_{1\le i\le nb}{h}_i$. We mention that two grids on the neighboring blocks $K_i$ can be non-matching on interface. We denote by ${\mathcal{T}}_h$, the partition of whole domain Ω, that is, ${\mathcal{T}}_h={\cup }_{i=1}^{nb}{\mathcal{T}}_h^i$. Then the discrete spaces for mixed formulation are defined by [49], [43], [13], [14]

$${\mathit{U}}_h^i\subset {\mathit{U}}_i,S_h^i\subset S_i,{\mathit{U}}_h={\oplus }_{i=1}^{nb}{\mathit{U}}_h^i,S_h={\oplus }_{i=1}^{nb}{S}_h^i,$$

where ${\mathit{U}}_h$ contains the polynomial of degree k and the normal components of vectors in ${\mathit{U}}_h$ are continuous between the elements on each $K_i$ but not across the interface.

The interface ${\gamma }_{ij}$ is then divided into quasi-uniform grids ${\mathcal{T}}_h^{ij}$. The mortar space on the interface ${\gamma }_{ij}$ containing the piece-wise polynomials of degree at least $k+1$, whether continuous or discontinuous, is denoted by ${\mathcal{L}}_h^{ij}\subset L^2({\gamma }_{ij})$. For a two dimensional domain, the interface has dimension one, and the polynomials of degree $k+1$ make up the interface space ${\mathcal{L}}_h^{ij}$. Let

$${\mathcal{L}}_h={\oplus }_{1\le i<j\le nb}{\mathcal{L}}_h^{ij},$$

be the corresponding global interface space. Note that, if the space is discontinuous then the mesh ${\mathcal{T}}_h^{ij}$ need not to be conforming. For each block $K_i$, we define the $L^2$ projection ${\mathcal{Q}}_h^i:L^2({\gamma }_i)⟶{\mathit{U}}_h^i\cdot {\mathit{\nu }}_i$ such that for any $\phi \in L^2({\gamma }_i)$

$${\langle \phi -{\mathcal{Q}}_h^i\phi ,\mathit{\psi }\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}=0,\mathit{\psi }\in {\mathit{U}}_h^i.$$ (2.1)

Projection ${\mathcal{Q}}_h^i$ maps the mortar space into the normal trace of velocity space on each block $K_i$.

Assumption 2.1

For the projection$\mathcal{Q}$defined in(2.1), there exists a constant C independent of h such that

$$\Vert m\Vert \le C[{\Vert {\mathcal{Q}}_h^{i}m\Vert }_{{\gamma }_{ij}}+{\Vert {\mathcal{Q}}_h^{j}m\Vert }_{{\gamma }_{ij}}],m\in {\mathcal{L}}_h,1\le i<j\le nb.$$ (2.2)

Condition (2.2) impose a limit on the mortar degree of freedom. In practice, this condition holds (cf. [54], [46]. Under the condition (2.2), our scheme (4.4) – (4.6) is solvable, accurate and stable. Here after any function $m\in {\mathcal{L}}_h$ is taken as extended by zero on ∂Ω. We mention that the partition ${\mathcal{T}}_h^{ij}$ do not need to be conforming if the space ${\mathcal{L}}_h^{ij}$ is discontinuous however conformity is necessary for error analysis.

3. Weakly continuous velocities and projection operators

First, we introduce some projection operators which will be used to compensate the analysis in sequence. We define the projection ${\mathcal{P}}_h:L^2(\gamma )⟶{\mathcal{L}}_h$ satisfying

$${\langle \psi -{\mathcal{P}}_h\psi ,m\rangle }_{\gamma }=0,\mathrm{\text{for any}}\psi \in L^2(\gamma )\mathrm{\text{and}}m\in {\mathcal{L}}_h.$$

Let $\pi :S⟶S_h$ be the orthogonal $L^2-\mathrm{\text{projection}}$ into $S_h$ defined by

$$(w-\pi w,\xi )=0,w\in S,\xi \in S_h.$$ (3.1)

Recall that

$$\mathrm{\nabla }\cdot {\mathit{U}}_h^i=S_h^i,$$

and there exists a projection operator ${\mathrm{\Pi }}_i$ of ${(H^{1/2+ϵ}(K_i))}^d\cap {\mathit{U}}_i$ onto ${\mathit{U}}_h^i$ (for any $ϵ>0$) satisfying

$${(\mathrm{\nabla }\cdot ({\mathrm{\Pi }}_i\mathit{\phi }-\mathit{\phi }),\xi )}_{K_i}=0,\xi \in S_h^i,{\langle (\mathit{\phi }-{\mathrm{\Pi }}_i\mathit{\phi })\cdot {\mathit{\nu }}_i,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i}=0,\mathit{v}\in {\mathit{U}}_h^i,$$ (3.2)

for any $\mathit{\phi }\in {(H^{1/2+ϵ}(K_i))}^d\cap {\mathit{U}}_i$. Observe that for any element face/edge e, $\mathit{\phi }\in (H^{1/2+ϵ}(K_i)),\mathit{\phi }\cdot \mathit{\nu }{|}_e\in H^{ϵ}(e)$; hence ${\mathrm{\Pi }}_i$ is well defined. Also the $H^{3/2+ϵ}$ regularity assumption ensures that we can apply ${\mathrm{\Pi }}_i$ to the flux arising from our elliptic problem. Mathew [39] provides that for any $\mathit{\phi }\in {(H^{ϵ}(K_i))}^d$, $0<ϵ<1$ with $\mathrm{\nabla }\cdot \mathit{\phi }=0$ then ${\mathrm{\Pi }}_i\mathit{\phi }$ is well defined and following approximation properties hold

$${\Vert {\mathrm{\Pi }}_i\mathit{\phi }\Vert }_{K_i}\le C{\Vert \mathit{\phi }\Vert }_{ϵ,K_i},{\Vert {\mathrm{\Pi }}_i\mathit{\phi }-\mathit{\phi }\Vert }_{K_i}\le C{h}^{ϵ}{\Vert \mathit{\phi }\Vert }_{ϵ,K_i}.$$

Here ${\Vert \cdot \Vert }_r$ is the $H^r-\mathrm{\text{norm}}$. We mention that the argument given in [39] for Raviart-Thomas spaces can be extended [30] for any of the mixed spaces under consideration to show that ${\mathrm{\Pi }}_i\mathit{\phi }$ is well defined and for any $\mathit{\phi }\in {(H^{ϵ}(K_i))}^d\cap {\mathit{U}}_i$

$${\Vert {\mathrm{\Pi }}_i\mathit{\phi }\Vert }_{K_i}\le C[{\Vert \mathit{\phi }\Vert }_{ϵ,K_i}+{\Vert \mathrm{\nabla }\cdot \mathit{\phi }\Vert }_{K_i}].$$

The above defined projection operators have the following approximation estimates [15], [18], [50]

$${\Vert \psi -{\mathcal{P}}_h\psi \Vert }_{-s,{\gamma }_{ij}}\le C{\Vert \psi \Vert }_{r,{\gamma }_{ij}}{h}^{r+s},0\le r\le k+2,0\le s\le k+2,$$ (3.3)

$$\Vert w-\pi w\Vert \le C{\Vert w\Vert }_r{h}^r,0\le r\le l+1,$$ (3.4)

$$\Vert \mathit{\phi }-{\mathrm{\Pi }}_i\mathit{\phi }\Vert \le C{\Vert \mathit{\phi }\Vert }_{r,K_i}{h}^r,1\le r\le k+1,$$ (3.5)

$${\Vert \mathrm{\nabla }\cdot (\mathit{\phi }-{\mathrm{\Pi }}_i\mathit{\phi })\Vert }_{K_i}\le C{\Vert \mathrm{\nabla }\cdot \mathit{\phi }\Vert }_{r,K_i}{h}^r,0\le r\le r+1,$$ (3.6)

$${\Vert \phi -{\mathcal{Q}}_h^i\phi \Vert }_{-s,{\gamma }_{ij}}\le C{\Vert \phi \Vert }_{r,{\gamma }_{ij}}{h}^{r+s},0\le r\le r+1,0\le s\le k+1,$$ (3.7)

$${\Vert \mathit{\phi }-{\mathrm{\Pi }}_i\mathit{\phi })\cdot {\mathit{\nu }}_i\Vert }_{-s,{\gamma }_{ij}}\le C{\Vert \mathit{\phi }\Vert }_{r,{\gamma }_{ij}}{h}^{r+s},0\le r\le k+1,0\le s\le k+1.$$ (3.8)

Note that Bound (3.3) – (3.4) and (3.6) – (3.8) are $L^2$ projection estimates [18] while approximations (3.5) can be found in [15], [50]. Here ${\Vert \cdot \Vert }_r$ denote the $H^r-\mathrm{\text{norm}}$ and ${\Vert \cdot \Vert }_{-s}$ is the $H^{-s}$ norm. The projection operator ${\mathrm{\Pi }}_i$ and ${\mathcal{Q}}_h^i$ satisfied

$$\mathrm{\nabla }\cdot {\mathrm{\Pi }}_i\mathit{\phi }=\pi \mathrm{\nabla }\cdot (\mathit{\phi }),({\mathrm{\Pi }}_i\mathit{\phi })\cdot {\mathit{\nu }}_i={\mathcal{Q}}_h^i(\mathit{\phi }\cdot {\mathit{\nu }}_i).$$

Following inequalities shall be used in the subsequent analysis

$${\Vert q\Vert }_{r,{\gamma }_{ij}}\le C{\Vert q\Vert }_{r+1/2,K_i},$$ (3.9)

$${\Vert \mathit{\phi }\cdot {\mathit{\nu }}_i\Vert }_{\partial K}\le C{h}^{-1/2}{\Vert \mathit{\phi }\Vert }_{K_i},$$ (3.10)

$${\langle q,\mathit{v}\cdot \mathit{\nu }\rangle }_{\partial K_i}\le {\Vert q\Vert }_{1/2,{\gamma }_{ij}}{\Vert \mathit{v}\Vert }_{\mathit{H}(div;K_i)}.$$ (3.11)

Bound (3.9) is the nonstandard trace theorem which can be found in (see [30], Theorem 1.5.2.1). The local inverse inequality (3.10) is given in [5]. Finally the bound (3.11) is proved in [15], [49]

For the purpose of analysis, we define the weakly continuous velocities [5] as

$${\mathit{U}}_h^c=\{\mathit{v}\in {\mathit{U}}_h:\sum _{i=1}^{nb}{\langle \mathit{v}{|}_{K_i}\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}=0,\mathrm{\text{for all}}m\in {\mathcal{L}}_h\}.$$ (3.12)

4. Numerical method

The weak formulation of the (1.3) – (1.5) is stated by; $\mathit{u}\in {\mathit{U}}_i,w\in W_i,\lambda \in \mathcal{L}$ such that

$${(\alpha (p)\mathit{u},\mathit{v})}_{K_i}={(p,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle \lambda ,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus \gamma },\mathit{v}\in {\mathit{U}}_i,$$ (4.1)

$${(\mathrm{\nabla }\cdot \mathit{u},w)}_{K_i}={(f(p),w)}_{K_i},w\in S_i,$$ (4.2)

$$\sum _{i=1}^{nb}{\langle \mathit{u}\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}=0,m\in \mathcal{L},$$ (4.3)

for $1\le i\le nb$. Here $\mathit{u}\cdot {\mathit{\nu }}_i$ denote the normal flux on each subdomain with ${\mathit{\nu }}_i$ outer unit normal to subdomain block $\partial K_i$. Equation (4.3) imposes the weak continuity of normal flux with respect to mortar space $\mathcal{L}$. Note that λ is the pressure on interface γ.

The mortar mixed approximation to the system (4.1) – (4.3) is read as; find ${\mathit{u}}_h\in {\mathit{U}}_h,p_h\in S_h,{\lambda }_h\in {\mathcal{L}}_h$ such that for $1\le i\le nb$

$${(\alpha (p_h){\mathit{u}}_h,\mathit{v})}_{K_i}={(p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle {\lambda }_h,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus {\gamma }_i},\mathit{v}\in {\mathit{U}}_h^i,$$ (4.4)

$${(\mathrm{\nabla }\cdot {\mathit{u}}_h,w)}_{K_i}={(f(p_h),w)}_{K_i}w\in S_h^i,$$ (4.5)

$$\sum _{i=1}^{nb}{\langle {\mathit{u}}_h\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}=0,m\in {\mathcal{L}}_h,$$ (4.6)

where last equation enforces flux continuity across mortar interface in weak sense with respect to mortar space. We shall employ mixed finite element method on each block $K_i$ to solve the problem locally. The unknown ${\lambda }_h$ approximate the mortar pressure p.

An application of continuous weak velocities (3.12) returns gloabl counter part of (4.4) – (4.6) which is read as; find ${\mathit{u}}_h\in {\mathit{U}}_h^c,p_h\in S_h$ satisfying

$$(\alpha (p_h){\mathit{u}}_h,\mathit{v})=\sum _{i=1}^{nb}{(p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial \mathrm{\Omega }},\mathit{v}\in {\mathit{U}}_h^c,$$ (4.7)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot {\mathit{u}}_h,w)}_{K_i}=(f(p_h),w),w\in S_h.$$ (4.8)

Under Hypothesis 2.1, the following lemma holds (cf. [5]).

Lemma 4.1

There exists a projection${\mathrm{\Pi }}_0:(H^{1/2+ϵ}(\mathrm{\Omega }))\cap \mathit{U}⟶{\mathit{U}}_h^c$such that

$${(\mathrm{\nabla }\cdot ({\mathrm{\Pi }}_0\mathit{\phi }-\mathit{\phi }),\tau )}_{\mathrm{\Omega }}=0,\tau \in S_h,$$

satisfying

$$\Vert {\mathrm{\Pi }}_0\mathit{\phi }-\mathrm{\Pi }\mathit{\phi }\Vert \le C\sum _{i=1}^{nb}{\Vert \mathit{\phi }\Vert }_{r+1/2,K_i}{h}^{r+1/2},0\le r\le k+1,\Vert {\mathrm{\Pi }}_0\mathit{\phi }-\mathit{\phi }\Vert \le C\sum _{i=1}^{nb}{\Vert \mathit{\phi }\Vert }_{r,K_i}{h}^r,1\le r\le k+1,$$

wherein $\mathrm{\Pi }\mathit{\phi }{|}_{K_i}={\mathrm{\Pi }}_i\mathit{\phi }$.

5. Existence of solution for discrete problem

We will demonstrate that the discrete problem has a unique solution in this section. We will apply the fixed-point argument (FPA) to establish the existence result. Prior to address existence, we will review the following version of Taylor's theorem. For $\rho \in S_h$ the relation

$$\alpha (\rho )-\alpha (p)=-\tilde{\alpha }(\rho )(p-\rho )=-{\alpha }_p(p)(p-\rho )+{\tilde{\alpha }}_{pp}(\rho ){(p-\rho )}^2.$$ (5.1)

holds. Here

$${\tilde{\alpha }}_p(\rho )=\underset{0}{\overset{1}{\int }}{\alpha }_p(\rho +t[p-\rho ])dt,{\tilde{\alpha }}_{pp}(\rho )=\underset{0}{\overset{1}{\int }}(1-t){\alpha }_{pp}(p+t[\rho -p])dt,$$

are bounded functions in $\overline{\mathrm{\Omega }}$. Also

$$f(\rho )-f(p)=-{\tilde{f}}_p(\rho )(p-\rho )=-f_p(p)(p-\rho )+{\tilde{f}}_{pp}(\rho ){(p-\rho )}^2.$$ (5.2)

Subtracting (4.7) - (4.8) from (4.1) – (4.2) and using (5.1), (5.2) we obtain the error equations

$${(\alpha (p)(\mathit{u}-{\mathit{u}}_h),\mathit{v})}_K=\sum _{i=1}^{nb}\{{(p-p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{(\tilde{\alpha }(p_h)(p-p_h){\mathit{u}}_h,\mathit{v})}_{K_i}-{\langle p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}\},\mathit{v}\in {\mathit{U}}_h^c,$$ (5.3)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot (\mathit{u}-{\mathit{u}}_h),w)}_{K_i}=({\tilde{f}}_p(p_h)(p-p_h),w),w\in S_h.$$ (5.4)

Further modification in (5.3) – (5.4) using (5.1) and (5.2) with $\rho =p_h$ leads to the system

$$(\alpha (p)(\mathit{u}-{\mathit{u}}_h),\mathit{v})-\sum _{i=1}^{nb}{(p-p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+(\mathbf{\Gamma }(p-p_h),\mathit{v})=({\tilde{\alpha }}_p(p_h)(p-p_h)(\mathit{u}-{\mathit{u}}_h),\mathit{v})+({\tilde{\alpha }}_{pp}(p_h){(p-p_h)}^2\mathit{u},\mathit{v})-\sum _{i=1}^{nb}{\langle p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},\mathit{v}\in {\mathit{U}}_h^c,$$ (5.5)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot (\mathit{u}-{\mathit{u}}_h),w)}_{K_i}-{(\beta (p-p_h),w)}_{K_i})=(-{\tilde{f}}_{pp}(p_h){(p-p_h)}^2,w),w\in S_h,$$ (5.6)

where $\mathbf{\Gamma }={\tilde{\alpha }}_p(p)\mathit{u}$ and $\beta =f_p(p)$. Making use of projection operators ${\mathrm{\Pi }}_i,\pi ,{\mathcal{P}}_h$, their properties (3.1), (3.2) and weakly continuous velocities, the system (5.5) – (5.6) returns

$$(\alpha (p)(\mathrm{\Pi }\mathit{u}-{\mathit{u}}_h),\mathit{v})-\sum _{i=1}^{nb}{(\pi p-p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+(\mathbf{\Gamma }(\pi p-p_h),\mathit{v})=(\alpha (p)(\mathrm{\Pi }\mathit{u}-\mathit{u}),\mathit{v})+(\mathbf{\Gamma }(\pi p-p),\mathit{v})+({\tilde{\alpha }}_p(p_h)(p-p_h)(\mathit{u}-{\mathit{u}}_h),\mathit{v})({\tilde{\alpha }}_{pp}(p_h){(p-p_h)}^2\mathit{u},\mathit{v})-\sum _{i=1}^{nb}{\langle p-{\mathcal{P}}_{h}p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},v\in {\mathit{U}}_h^c,\sum _{i=1}^{nb}(\mathrm{\nabla }\cdot (\mathrm{\Pi }\mathit{u}-{\mathit{u}}_h),w)-(\beta (\pi p-p_h),w)=(\beta (p-\pi p),w)+(-{\tilde{f}}_{pp}(p_h){(p-p_h)}^2,w),w\in S_h.$$

Define the operator $J:H^2(\mathrm{\Omega })⟶L^2(\mathrm{\Omega })$ by

$$J\omega =-\mathrm{\nabla }\cdot (\mathcal{A}(p)\mathrm{\nabla }\omega +\mathcal{A}(p)\mathbf{\Gamma }\omega )-\beta \omega .$$

The adjoint $J^{⁎}$ of the operator J is defined by

$$J^{⁎}\chi =-\mathrm{\nabla }\cdot (\mathcal{A}(p)\mathrm{\nabla }\chi )+\mathcal{A}(p)\mathbf{\Gamma }\mathrm{\nabla }\chi -\beta \chi .$$

We assume that the restrictions of operators J and $J^{⁎}$ to $H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ have bounded inverses; that is, there exists a unique $\varphi \in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ such that $J\varphi =\psi$ (respectively $J^{⁎}\varphi =\psi$) for any $\psi \in L^2(\mathrm{\Omega })$ and

$${\Vert \varphi \Vert }_2\le C\Vert \psi \Vert .$$

Let the map $\mathrm{\Phi }:{\mathit{U}}_h^c\times S_h⟶{\mathit{U}}_h^c\times S_h$ be given by

$$\mathrm{\Phi }(\mathit{\eta },\rho )=(\mathit{y},z);\mathrm{\text{where}}(\mathit{y},z)\mathrm{\text{is the solution of the system}}$$

$$\begin{gathered} \sum _{i=1}^{nb}\{{(\alpha (p)({\mathrm{\Pi }}_i\mathit{u}-\mathit{y}),\mathit{v})}_{K_i}-{(\pi p-z,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{(\mathbf{\Gamma }(\pi p-z),\mathit{v})}_{K_i}\}=\sum _{i=1}^{nb}{\{\alpha (p)({\mathrm{\Pi }}_i\mathit{u}-\mathit{u}),\mathit{v})}_{K_i}+{(\mathbf{\Gamma }(\pi p-p),\mathit{v})}_{K_i} \\ +{({\tilde{\alpha }}_p(\rho )(p-\rho )(\mathit{u}-\mathit{\eta }),\mathit{v})}_{K_i}+{({\tilde{\alpha }}_{pp}(\rho ){(p-\rho )}^2,\mathit{v})}_{K_i} \\ -{\langle p-{\mathcal{P}}_{h}p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}\}\mathit{v}\in {\mathit{U}}_h^c, \end{gathered}$$ (5.7)

$$\begin{gathered} \sum _{i=1}^{nb}\{{(\mathrm{\nabla }\cdot ({\mathrm{\Pi }}_i\mathit{u}-\mathit{y}),w)}_{K_i}-{(\beta (\pi p-z),w)}_{K_i} \\ =\sum _{i=1}^{nb}{\{\beta (p-\pi p),w)}_{K_i}+{(-{\tilde{f}}_{pp}(\rho ){(p-\rho )}^2,w)}_{K_i}\},w\in S_h. \end{gathered}$$ (5.8)

Observe that the system (5.7) – (5.8) accord with the mixed method for linear operator J. Thus existence of (5.7) – (5.8) for small h follows from [5] with [19]. Hence $(\mathit{y},z)$ is the solution of the system of the form

$$(\alpha (p)\mathit{\psi },\mathit{v})-(\phi ,\mathrm{\nabla }\cdot \mathit{v})+(\mathbf{\Gamma }\phi ,\mathit{v})=F(\mathit{v}),\mathit{v}\in {\mathit{U}}_h^c,$$ (5.9)

$$(\mathrm{\nabla }\cdot \mathit{\psi },w)-(\beta \phi ,w)=G(w),w\in S_h.$$ (5.10)

Problem (5.9) - (5.10) has unique solution $(\mathit{\psi },\phi )\in {\mathit{U}}_h^c\times S_h$ for sufficiently small h and any $F\in {\mathit{U}}^{\prime },G\in S^{\prime }$ where existence follows from uniqueness. Consequently, the existence of solution for $({\mathit{u}}_h,p_h)\in {\mathit{U}}_h^c\times S_h$ for the system (4.7) – (4.8) is similar to show that the map Φ has a fixed point. Therefore, the existence of solution for (4.7) – (4.8) follows from Brouwer fixed point theorem if we prove that Φ maps a ball of ${\mathit{U}}_h^c\times S_h$ to itself. For this purpose, we need the following lemma.

Lemma 5.1

Let$2\le \theta <\infty$. Let$\mathit{\omega }\in {\mathit{U}}_h^c$,$\mathit{q}\in {(L^2(\mathrm{\Omega }))}^2,r\in L^2(\mathrm{\Omega })$and$\tilde{q}\in L^2(\gamma )$, if$\tau \in S_h$satisfies

$$\sum _{i=1}^{nb}\{{(\alpha (p)\mathit{\omega },\mathit{v})}_{K_i}-{(\tau ,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{(\mathbf{\Gamma }\tau ,\mathit{v})}_{K_i}\}=\sum _{i=1}^{nb}\{{(\mathit{q},\mathit{v})}_{K_i}+{\langle \tilde{q},\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}\}\mathit{v}\in {\mathit{U}}_h^c,$$ (5.11)

$$\sum _{i=1}^{nb}\{{(\mathrm{\nabla }\cdot \mathit{\omega },w)}_{K_i}-{(\beta \tau ,w)}_{K_i}\}=\sum _{i=1}^{nb}{(r,w)}_{K_i}w\in S_h,$$ (5.12)

then there exists a constant C such that

$${\Vert \tau \Vert }_{0,\theta }\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert \mathit{\omega }\Vert }_{K_i}+h^{1+2/\theta (1-{\delta }_{0k})}{\Vert \mathrm{\nabla }\cdot \mathit{\omega }\Vert }_{K_i}+{\Vert \mathit{q}\Vert }_{K_i}+{\Vert r\Vert }_{K_i}+{\Vert \tilde{q}\Vert }_{1/2,{\gamma }_i}],$$ (5.13)

for sufficiently small h where${\delta }_{ij}$denote the Kronecker symbol.

Proof

Let ${\theta }^{\prime }$ be the conjugate exponent of θ. Let $\phi \in W^{2,{\theta }^{\prime }}(\mathrm{\Omega })$ be the unique solution of adjoint problem

$$\begin{gathered} J^{⁎}\phi =\psi ,\mathrm{\text{in}}\mathrm{\Omega } \\ \phi =0,\mathrm{\text{on}}\partial \mathrm{\Omega }, \end{gathered}$$

satisfying the elliptic regularity estimate

$${\Vert \phi \Vert }_{2,{\theta }^{\prime }}\le C{\Vert \psi \Vert }_{0,{\theta }^{\prime }}.$$

Now

$$(\tau ,\psi )=(\tau ,J^{⁎}\phi )=\sum _{i=1}^{nb}{(\tau ,-\mathrm{\nabla }\cdot (\mathcal{A}(p)\mathrm{\nabla }\phi )+\mathcal{A}(p)\mathbf{\Gamma }\phi -\beta \phi )}_{K_i}.$$

Using (3.1), (3.2), (5.11), (5.12) and integration by part, we have the relation

$$\begin{gathered} (\tau ,\psi )=\sum _{i=1}^{nb}\{{(\mathit{q},\mathcal{A}(p)\mathrm{\nabla }\phi )}_{K_i}+{(\mathit{q},{\mathrm{\Pi }}_i\mathcal{A}(p)\mathrm{\nabla }\phi -\mathcal{A}(p)\mathrm{\nabla }\phi )}_{K_i} \\ +{(\alpha (p)\mathit{\omega },\mathcal{A}(p)\mathrm{\nabla }\phi -{\mathrm{\Pi }}_i\mathcal{A}(p)\mathrm{\nabla }\phi )}_{K_i}\}+{(\mathbf{\Gamma }\tau ,\mathcal{A}(p)\mathrm{\nabla }\phi -{\mathrm{\Pi }}_i\mathcal{A}(p)\mathrm{\nabla }\phi )}_{K_i} \\ {(\mathrm{\nabla }\cdot \mathit{\omega },\phi -P_h\phi )}_{K_i}+{(\beta \tau ,P_h\phi -\phi )}_{K_i}+{(r,P_h\phi -\phi )}_{K_i} \\ {(r,\phi )}_{K_i}+{\langle \tilde{q},{\mathrm{\Pi }}_i\mathcal{A}(p)\mathrm{\nabla }\phi \cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}. \end{gathered}$$

Then the estimate (5.13) follows from [40], [8]. □

We shall now address our main theorem regarding existence of solution of discrete problem. Let ${\mathit{U}}_h={\mathit{U}}_h$ and ${\mathcal{S}}_h=S_h$ with stronger norm ${\Vert \mathit{v}\Vert }_{{\mathit{U}}_h}={\Vert \mathit{v}\Vert }_{0,2+ϵ}+\Vert div\mathit{v}\Vert$ and $\Vert w\Vert ={\Vert w\Vert }_{0,(4+2ϵ)/ϵ}$ respectively. The solvability of discrete system follows from the following theorem.

Theorem 5.1

For sufficiently small$\delta >0$, the map Φ maps a ball of radius δ of ${\mathit{U}}_h\times {\mathcal{S}}_h$ into itself.

Proof

Let $\theta =\frac{4+2ϵ}{ϵ}>4$ so that $\frac{1}{\theta }+\frac{1}{2+ϵ}=\frac{1}{2}$. Note that $ϵ<1$ implies $\theta >4$. Let

$$\Vert \mathrm{\Pi }\mathit{u}-\mathit{\eta }\Vert \le \delta ,\mathrm{\text{and}}\Vert \pi p-\rho \Vert \le \delta <1.$$

Applying Lemma 5.1 to (5.7) - (5.8) with

$$\mathit{\omega }=\mathrm{\Pi }\mathit{u}-\mathit{y},\tau =\pi p-z,q=\alpha (p)(\mathrm{\Pi }\mathit{u}-\mathit{u})+\mathbf{\Gamma }(\pi p-p)+{\tilde{\alpha }}_p(\rho )(p-\rho )(\mathit{u}-\mathit{\eta })+{\tilde{\alpha }}_{pp}(\rho ){(p-\rho )}^2,r=\beta (p-\pi p)-{\tilde{f}}_{pp}(\rho ){(p-\rho )}^2,\tilde{q}=p-{\mathcal{P}}_{h}p,$$

and using (3.5), (3.4), (3.3) and (3.9) and Sobolev imbedding inequalities [2], we see that

$$\begin{gathered} {\Vert \tau \Vert }_{0,\theta }\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{K_i}+h^{1+2/\theta (1-{\delta }_{0k})}{\Vert \mathrm{\nabla }\cdot (\mathrm{\Pi }\mathit{u}-\mathit{y})\Vert }_{K_i} \\ +\Vert \alpha (p){\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{u}\Vert }_{K_i}+{\Vert \mathbf{\Gamma }(\pi p-p)\Vert }_{K_i}+{\Vert {\tilde{\alpha }}_p(p_h)(p-\rho )(\mathit{u}-\mathit{\eta })\Vert }_{K_i} \\ +{\Vert {\tilde{\alpha }}_{pp}(p_h){(p-\rho )}^2\Vert }_{K_i}+{\Vert \beta (p-\pi p)\Vert }_{K_i}+{\Vert {\tilde{f}}_{pp}(p_h){(p-\rho )}^2\Vert }_{K_i} \\ +{\Vert p-{\mathcal{P}}_{h}p\Vert }_{1/2,{\gamma }_i}]\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}+h{\Vert p\Vert }_{2,K_i}+{\Vert p-\rho \Vert }_{0,4,K_i}^2 \\ +{\Vert p-\rho \Vert }_{0,\theta ,K_i}{\Vert \mathit{u}-\mathit{\eta }\Vert }_{0,2+ϵ,K_i}+{\Vert p-{\mathcal{P}}_{h}p\Vert }_{1/2,K_i}]\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}+h{\Vert p\Vert }_{2,K_i}+{\Vert p-\pi p\Vert }_{0,\theta ,K_i}^2+{\Vert \pi p-\rho \Vert }_{0,\theta ,K_i}^2 \\ +({\Vert p-\pi p\Vert }_{0,\theta ,K_i}+{\Vert \pi p-\rho \Vert }_{0,\theta ,K_i})({\Vert \mathit{u}-{\mathrm{\Pi }}_i\mathit{u}\Vert }_{0,2+ϵ,K_i}+{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{\eta }\Vert }_{0,2+ϵ,K_i})\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}+h{\Vert p\Vert }_{2,K_i}+{\delta }^2+(h{\Vert p\Vert }_{1,\theta ,K_i}+\delta )(h{\Vert \mathit{u}\Vert }_{1,2+ϵ,K_i}+\delta )]\le C\sum _{i=1}^{nb}[h^{2/\theta }{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}+(h+{\delta }^2){\Vert p\Vert }_{2+ϵ}^2]. \end{gathered}$$ (5.14)

If we move the last term in (5.7) to the right, making left hand side of (5.7) – (5.8) mixed method for the operator $-\mathrm{\nabla }\cdot (\mathcal{A}(p)\mathrm{\nabla }\cdot )$, then we have the estimate

$${\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}\le C\sum _{i=1}^{nb}[{\Vert \pi p-z\Vert }_{K_i}+{\Vert \mathit{q}\Vert }_{K_i}+{\Vert r\Vert }_{K_i}+{\Vert \tilde{q}\Vert }_{1/2,K_i}]\le C\sum _{i=1}^{nb}{[\pi p-z\Vert }_{0,\theta }+(h+{\delta }^2)].$$ (5.15)

Combining (5.15) and (5.14), we obtain

$${\Vert \pi p-z\Vert }_{0,\theta }\le C[h^{\theta /2}{\Vert \pi p-z\Vert }_{0,\theta ,K_i}+(h+{\delta }^2)].$$ (5.16)

For sufficiently small h, we have from (5.16)

$${\Vert \pi p-z\Vert }_{0,\theta }\le C_1[h+{\delta }^2].$$ (5.17)

A combination of (5.16) and (5.17) yield

$${\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}\le C_2[h+{\delta }^2].$$ (5.18)

Since the mesh under consideration is quasi-uniform so inverse estimate and (5.18) implies

$${\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{0,2+ϵ}\le C_{ϵ}{h}^{-ϵ/(2+ϵ)}\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert \le C_{ϵ}{h}^{-ϵ/(2+ϵ)}{C}_2[h+{\delta }^2]\le C_3[h^{2/(2+ϵ)}+{\delta }^2{h}^{-ϵ/(2+ϵ)}].$$ (5.19)

Adding (5.18) and (5.19), we see that

$${\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}\le 2C_3[h^{2/(2+ϵ)}+{\delta }^2{h}^{-ϵ/(2+ϵ)}].$$ (5.20)

Choosing $\delta =4C_3{h}^{2/(2+ϵ)}$ then equation (5.20) suggests that $2C_3{h}^{2/(2+ϵ)}\le \delta /2$ and $2C_3{\delta }^2{h}^{-ϵ/(2+ϵ)}\le \delta /2$. Thus we must have $\delta \in [{(4C_3)}^{-1}{h}^{ϵ/(2+ϵ)},4C_3{h}^{ϵ/(2+ϵ)}]\ne \varnothing$. Therefore, (5.17) and (5.20) imply

$${\Vert \pi p-z\Vert }_{0,\theta }\le \delta ,{\Vert \mathrm{\Pi }\mathit{u}-\mathit{y}\Vert }_{\mathit{U}}\le \delta .$$ (5.21)

Hence Φ maps the ball of radius δ centered at $(\mathrm{\Pi }\mathit{u},\pi p)$ onto itself. Existence is established. □

6. Error estimates

This section is dedicated to a priori error estimates for proposed method. In ordered to derive error estimates, we need the following bounds (cf. [40]).

Corollary 6.1

For$0<ϵ<1$and$\theta =(4+2ϵ)/ϵ$, there exists a sequence of discrete solutions$\{{\mathit{u}}_h,p_h\}⟶\{\mathit{u},p\},h⟶0$satisfying

$$\max \{{\Vert \mathit{u}-{\mathit{u}}_h\Vert }_{0,2+ϵ},{\Vert p-p_h\Vert }_{0,\theta }\}\le C{h}^{2/(2+ϵ)},$$

Moreover,

$${\Vert {\mathit{u}}_h\Vert }_{0,\infty }\le C(1+{\Vert p\Vert }_{2+ϵ}^2).$$ (6.1)

Using first order Taylor expansion, the error equations (5.5) – (5.6) can be written in the form

$$\begin{gathered} (\alpha (p)(\mathit{u}-{\mathit{u}}_h),\mathit{v})-\sum _{i=1}^{nb}{(p-p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{\langle p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i} \\ +({\tilde{\alpha }}_p(p_h)(p-p_h){\mathit{u}}_h,\mathit{v})=0,\mathit{v}\in {\mathit{U}}_h^c, \end{gathered}$$ (6.2)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot (\mathit{u}-{\mathit{u}}_h),w)}_{K_i}=({\tilde{f}}_p(p_h)(p-p_h),w),w\in S_h,$$ (6.3)

where $\widetilde{{\alpha }_p}$ and ${\tilde{f}}_p$ are bounded functions.

Note that the mixed finite element method for the operator $L:H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })⟶L^2(\mathrm{\Omega })$ define

$$L\omega =-\mathrm{\nabla }\cdot (\mathcal{A}(\mathit{x},p)\mathrm{\nabla }\omega +\mathcal{A}(\mathit{x},p)[{\tilde{\alpha }}_p(p_h){\mathit{u}}_h]\omega ),$$

with formal adjoint

$$L^{⁎}\chi =-\mathrm{\nabla }\cdot (\mathcal{A}(\mathit{x},p)\mathrm{\nabla }\chi )+\mathcal{A}(\mathit{x},p)[{\tilde{\alpha }}_p(p_h){\mathit{u}}_h]\mathrm{\nabla }\chi ,$$

corresponds to the error equations (6.2) – (6.3). Following technical lemma is required to apply the duality argument in the derivation of convergence estimates.

Lemma 6.1

There exists$h_0>0$such that if$h<h_0$. The operator$L^{⁎}$has a bounded inverse mapping from$H^2(\mathrm{\Omega })$onto$H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$.

Theorem 6.1

If condition(2.2)holds, then for solutions$\{{\mathit{u}}_h,p_h\}$of the system(4.4)–(4.6), there exists a positive constant C independent of h such that

$$\Vert \mathrm{\nabla }\cdot (\mathit{u}-{\mathit{u}}_h)\Vert \le C\sum _{i=1}^{nb}[{\Vert \mathit{u}\Vert }_{r,K_i}+{\Vert \mathrm{\nabla }\cdot \mathit{u}\Vert }_{r,K_i}+{\Vert p\Vert }_{r+1,K_i}]h^r,1\le r\le k+1.\Vert \mathit{u}-{\mathit{u}}_h\Vert \le C\sum _{i=1}^{nb}[{\Vert \mathit{u}\Vert }_{r,K_i}+{\Vert p\Vert }_{r+1,K_i}]h^r,1\le r\le k+1.\Vert p-p_h\Vert \le C\sum _{i=1}^{nb}[{\Vert \mathit{u}\Vert }_{r,K_i}+{\Vert p\Vert }_{r+1,K_i}]h^r,1\le r\le \min \{k+1,l+1\}.$$

Proof

For convenience, we introduce the notations

$$\mathit{d}=\mathit{u}-{\mathit{u}}_h,\mathit{e}={\mathrm{\Pi }}_i\mathit{u}-{\mathit{u}}_h,\mathrm{\Lambda }=p-p_h,\tau =\pi p-p.$$

Rewrite the system (6.2) – (6.3) and using above notations, we arrived at

$$\begin{gathered} \sum _{i=1}^{nb}\{{(\alpha (p)\mathit{e},\mathit{v})}_{K_i}-{(\tau ,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{([\tilde{\alpha }(p_h){\mathit{u}}_h]\tau ,\mathit{v})}_{K_i}\}=\sum _{i=1}^{nb}\{{(\alpha (p)(\mathrm{\Pi }\mathit{u}-\mathit{u}),\mathit{v})}_{K_i}+{([{\tilde{\alpha }}_p(p_h){\mathit{u}}_h](\pi p-p),\mathit{v})}_{K_i} \\ +{\langle \mathcal{P}p-p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}\},\mathit{v}\in {\mathit{U}}_h^0, \end{gathered}$$ (6.4)

$$\sum _{i=1}^{nb}\{{(\mathrm{\nabla }\cdot \mathit{e},w)}_{K_i}-{({\tilde{f}}_p(p_h)\tau ,w)}_{K_i}\}={({\tilde{f}}_p(p_h)(p-\pi p),w)}_{K_i},w\in S_h.$$ (6.5)

Applying Lemma 5.1 to the system (6.4) – (6.5) with condition $\theta =2$ combined with Lemma 6.1 and utilizing (3.3), (3.4), (3.5), (3.9), and (6.1), we observe that

$$\Vert \tau \Vert \le \sum _{i=1}^{nb}[h{\Vert \mathit{e}\Vert }_{K_i}+h^{2-{\delta }_{0k}}{\Vert \mathrm{\nabla }\cdot \mathit{e}\Vert }_{K_i}+{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{u}\Vert }_{K_i}+{\Vert \pi p-p\Vert }_{K_i}+{\Vert \mathcal{P}p-p\Vert }_{1/2,{\gamma }_i}]\le C\sum _{i=1}^{nb}[h{\Vert \mathit{e}\Vert }_{K_i}+h^{2-{\delta }_{0k}}{\Vert \mathrm{\nabla }\cdot \mathit{e}\Vert }_{K_i}+h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h^r{\Vert p\Vert }_{r+1}].$$ (6.6)

Now using ${\mathrm{\Pi }}_i$ and π, we rewrite (5.5) – (5.6) in the form

$$\begin{gathered} \sum _{i=1}^{nb}\{{(\alpha (p)\mathit{e},\mathit{v})}_{K_i}-{(\tau ,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}\}=\sum _{i=1}^{nb}\{{({\mathrm{\Pi }}_i\mathit{u}-\mathit{u},\mathit{v})}_{K_i}-{({\tilde{\alpha }}_p(p_h){\mathit{u}}_h\mathrm{\Lambda },\mathit{v})}_{K_i} \\ +{\langle p-{\mathcal{P}}_{h}p,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}\},\mathit{v}\in {\mathit{U}}_h^0,\sum _{i=1}^{nb}(\mathrm{\nabla }\cdot \mathit{e},w)=({\tilde{f}}_p(p_h)\mathrm{\Lambda },w),w\in W_h, \end{gathered}$$

then the estimate

$${\Vert \mathit{e}\Vert }_{\mathit{U}}\le C\sum _{i=1}^{nb}[{\Vert {\mathrm{\Pi }}_i\mathit{u}-\mathit{u}\Vert }_{K_i}+{\Vert {\mathit{u}}_h\Vert }_{0,\infty }{\Vert \mathrm{\Lambda }\Vert }_{K_i}+{\Vert \mathrm{\Lambda }\Vert }_{K_i}+{\Vert p-{\mathcal{P}}_{h}p\Vert }_{1/2,{\gamma }_i}],$$ (6.7)

is obtained from (6.1) [12] (adopting the same procedure as did to obtain (5.17)). Using (3.3) (3.5) and (3.9), (6.7) implies

$${\Vert \mathit{e}\Vert }_{\mathit{U}}\le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+{\Vert \mathrm{\Lambda }\Vert }_{K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$ (6.8)

Substituting (6.8) in (6.6), we see that

$$\Vert \tau \Vert \le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h{\Vert \mathrm{\Lambda }\Vert }_{K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$ (6.9)

Combining triangle inequality with (6.9) and (3.4), we arrived at

$$\Vert \mathrm{\Lambda }\Vert =\Vert p-p_h\Vert \le \Vert p-\pi p\Vert +\Vert \tau \Vert \le C\sum _{i=1}^{nb}[h^r{\Vert p\Vert }_{r,K_i}+h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h{\Vert \mathrm{\Lambda }\Vert }_{K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$

For sufficiently small h, the above relation implies that

$$\Vert \mathrm{\Lambda }\Vert \le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$ (6.10)

Put back (6.10) in (6.8)

$${\Vert \mathit{e}\Vert }_{\mathit{U}}\le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$ (6.11)

Applying triangle inequality and using (6.11) and (3.5), we see that

$$\Vert \mathit{d}\Vert \le \Vert \mathit{e}\Vert +\Vert \mathit{u}-\mathrm{\Pi }\mathit{u}\Vert \le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$ (6.12)

Finally using (6.11) and (3.6), we obtain

$$\Vert \mathrm{\nabla }\cdot \mathit{d}\Vert \le \sum _{i=1}^{nb}[{\Vert \mathrm{\nabla }\cdot \mathit{e}\Vert }_{K_i}+{\Vert \mathrm{\nabla }\cdot (\mathit{u}-\mathrm{\Pi }\mathit{u})\Vert }_{K_i}].\le C\sum _{i=1}^{nb}[h^r{\Vert \mathit{u}\Vert }_{r,K_i}+h^r{\Vert \mathrm{\nabla }\cdot \mathit{u}\Vert }_{r,K_i}+h^r{\Vert p\Vert }_{r+1,K_i}].$$

 □

7. Uniqueness

Let ${\{{\mathit{u}}_h^i,p_h^i\}}_{i=1}^2\in {\mathit{U}}_h^c\times S_h$ be the solution of (4.7) – (4.8) satisfying Theorem 6.1 provided that they satisfy (5.21). Then the system (4.7) – (4.8) implies that

$$(\alpha (p_h^1){\mathit{u}}_h^1,\mathit{v})=\sum _{i=1}^{nb}{(p_h^1,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial \mathrm{\Omega }},\mathit{v}\in {\mathit{U}}_h^c,$$ (7.1)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot {\mathit{u}}_h^1,w)}_{K_i}=(f(p_h^1),w),w\in S_h,$$ (7.2)

and

$${(\alpha (p_h^2){\mathit{u}}_h^2,\mathit{v})}_{K_i}=\sum _{i=1}^{nb}{(p_h^2,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial \mathrm{\Omega }},\mathit{v}\in {\mathit{U}}_h^c,$$ (7.3)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot {\mathit{u}}_h^2,w)}_{K_i}=(f(p_h^2),w),w\in S_h.$$ (7.4)

Subtracting (7.3) – (7.4) from (7.1) – (7.2), we see that

$$(\alpha (p_h^1){\mathit{u}}_h^1-\alpha (p_h^2){\mathit{u}}_h^2,\mathit{v})=\sum _{i=1}^{nb}{(p_h^1-p_h^2,\mathrm{\nabla }\cdot \mathit{v})}_{K_i},\mathit{v}\in {\mathit{U}}_h^c,$$ (7.5)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot ({\mathit{u}}_h^1-{\mathit{u}}_h^2),w)}_{K_i}=(f(p_h^1)-f(p_h^2),w),w\in S_h.$$ (7.6)

Using (5.1) and (5.2), we get

$$\alpha (p_h^1){\mathit{u}}_h^1-\alpha (p_h^2){\mathit{u}}_h^2=\alpha (p_h^1)({\mathit{u}}_h^1-{\mathit{u}}_h^2)+{\tilde{\alpha }}_p(p_h^1)(p_h^1-p_h^2){\mathit{u}}_h^2.$$ (7.7)

$$f(p_h^1)-f(p_h^2)=-{\tilde{f}}_p(p_h^1)(p_h^2-p_h^1)={\tilde{f}}_p(p_h^1)(p_h^1-p_h^2).$$ (7.8)

Substituting (7.7) and (7.8) in (7.5) - (7.6), and using the notations

$$P=p_h^1-p_h^2,Q={\mathit{u}}_h^1-{\mathit{u}}_h^2,$$

we see that

$$\sum _{i=1}^{nb}\{{(\alpha (p_h^1)\mathit{Q},\mathit{v})}_{K_i}-{(P,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{({\tilde{\alpha }}_p(p_h^1){\mathit{u}}_h^{2}P,\mathit{v})}_{K_i}\}=0,\mathit{v}\in {\mathit{U}}_h^c,$$ (7.9)

$$\sum _{i=1}^{nb}{(\mathrm{\nabla }\cdot \mathit{Q},w)}_{K_i}-{({\tilde{f}}_p(p_h^1),w)}_{K_i}=0,w\in S_h.$$ (7.10)

Then we have the following bound (cf. [19])

$$\Vert \mathit{Q}\Vert \le C\Vert P\Vert ,$$ (7.11)

$$\Vert \mathrm{\nabla }\cdot \mathit{Q}\Vert \le C\Vert P\Vert .$$ (7.12)

Now modifying the term $\alpha (p_h^1)\mathit{Q}$, we can rewrite the system (7.9) – (7.10) in the form

$$\sum _{i=1}^{nb}{\{\alpha (p)\mathit{Q},\mathit{v})}_{K_i}-{(P,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{({\tilde{\alpha }}_p(p_h^1){\mathit{u}}_h^{2}P,\mathit{v})}_{K_i}\}=\sum _{i=1}^{nb}{({\tilde{\alpha }}_p(p_h^1)(p-p_h^1)\mathit{Q},\mathit{v})}_{K_i},\mathit{v}\in {\mathit{U}}_h^c,$$ (7.13)

$$\sum _{i=1}^{nb}\{{(\mathrm{\nabla }\cdot \mathit{Q},w)}_{K_i}-{({\tilde{f}}_p(p_h^1)P,w)}_{K_i}=0\},w\in S_h.$$ (7.14)

Applying Lemma 5.1 to (7.13) – (7.14) with $\theta =2$, we see that

$$\Vert P\Vert \le C\sum _{i=1}^{nb}[h({\Vert \mathit{Q}\Vert }_{K_i}+{\Vert \mathrm{\nabla }\cdot \mathit{Q}\Vert }_{K_i})].$$ (7.15)

Combining (7.11), (7.12) and (7.15), we have

$$\Vert P\Vert \le Ch\Vert P\Vert ,$$ (7.16)

which in turn implies that, for sufficiently small h, $\Vert P\Vert \le 0$, that is, $P=0$. Hence (7.11) gives $\mathit{Q}=0$. Thus $p_h^1=p_h^2$ and ${\mathit{u}}_h^1={\mathit{u}}_h^2$.

8. Mortar interface formulation

We now turn our attention to the mortar interface formulations and present the error estimate for the approximation to the mortar pressure. The accuracy of the data transmission between the subdomains is indicated by the order of convergence in interface space. For this, let us define the pressure dependent bilinear form $D_h=L^2(\gamma )\times L^2(\gamma )⟶\mathit{R}$ by

$$D_h(\lambda ,m)=\sum _{i=1}^{nb}{D}_{h,i}(\lambda ,m)=-\sum _{i=1}^{nb}{\langle {\mathit{u}}_h^{⁎}(\lambda )\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i},\lambda ,m\in L^2(\gamma ).$$

where ${\mathit{u}}_h^{⁎}(\lambda )\in {\mathit{U}}_h\times S_h$ is a component of solution $({\mathit{u}}_h^{⁎}(\lambda ),p_h^{⁎}(\lambda ))$ solves the local problem

$${(\alpha (p_h){\mathit{u}}_h^{⁎}(\lambda ),\mathit{v})}_{K_i}={(p_h^{⁎}(\lambda ),\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle \lambda ,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},\mathit{v}\in {\mathit{U}}_h^i,{(\mathrm{\nabla }\cdot \cdot {\mathit{u}}_h^{⁎},w)}_{K_i}=0,w\in S_h^i,$$

for $1\le i\le nb$ with given λ and $f=0,g=0$.

Next, we define the linear functional $G_h:L^2(\gamma )⟶\mathit{R}$ by

$$G_h=\sum _{i=1}^{nb}{G}_{h,i}(m)=\sum _{i=1}^{nb}{\langle {\overline{\mathit{u}}}_h,\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i},$$

where $({\overline{\mathit{u}}}_h,{\overline{p}}_h)\in {\mathit{U}}_h\times S_h$ solves the problem

$${(\alpha (p_h){\overline{\mathit{u}}}_h,\mathit{v})}_{K_i}={({\overline{p}}_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle g,\mathit{v}\cdot \mathit{\nu }\rangle }_{\partial K_i\setminus \gamma },\mathit{v}\in {\mathit{U}}_h^i,{(\mathrm{\nabla }\cdot {\overline{\mathit{u}}}_h,w)}_{K_i}={(f(p_h),w)}_{K_i},w\in S_h^i,$$

for $1\le i\le nb$ with $\lambda =0$ and given f and g.

It is easy to show [29] that the solution of the discrete system (4.4)-(4.6) satisfies the problem

$$D_h({\lambda }_h,m)=G_h(m),m\in {\mathcal{L}}_h,$$

with

$${\mathit{u}}_h={\mathit{u}}_h^{⁎}({\lambda }_h)+{\overline{\mathit{u}}}_h,p_h=p_h^{⁎}({\lambda }_h)+{\overline{p}}_h.$$

Lemma 8.1

The interface bilinear form$D_h(\cdot ,\cdot )$is symmetric and positive semi-definite on$L^2(\gamma )$. If condition(2.2)holds, then$D_h(\cdot ,\cdot )$is positive definite on${\mathcal{L}}_h$. Moreover

$$D_{h,i}(m,m)={(\alpha (p_h){\mathit{u}}_h^{⁎}(m),{\mathit{u}}_h^{⁎}(m))}_{K_i}\ge 0.$$ (8.1)

Let us define the semi-norm induced by the bilinear form $D_h(\cdot ,\cdot )$ on $L^2(\gamma )$ by

$${\Vert m\Vert }_{D_h}=\sqrt{D_h(m,m)},m\in L^2(\gamma ).$$

Theorem 8.1

I f condition(2.2)holds, then for the interface pressure${\lambda }_h$there exists a positive constant C such that

$$\Vert p-{\lambda }_h\Vert \le \sum _{i=1}^{nb}[{\Vert p\Vert }_{r+1}+\Vert \mathit{u}-{\mathit{u}}_h\Vert ]h^r.$$

Proof

Note that $({\mathit{u}}_h(m),p_h(m))\in {\mathit{U}}_h\times W_h$ satisfy

$${(\alpha (p_h){\mathit{u}}_h(m),\mathit{v})}_{K_i}={(p_h(m),\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus \gamma },\mathit{v}\in {\mathit{U}}_{h,i},$$ (8.2)

$${(\mathrm{\nabla }\cdot {\mathit{u}}_h(m),w)}_{K_i}={(f,w)}_{K_i},w\in S_{h,i},$$ (8.3)

with

$${\mathit{u}}_h(m)={\mathit{u}}_h^{⁎}(m)+\overline{\mathit{u}},p_h(m)=p_h^{⁎}(m)+{\overline{p}}_h,m\in L^2(\gamma ).$$

Particularly, ${\mathit{u}}_h({\lambda }_h)={\mathit{u}}_h$ and $p_h({\lambda }_h)=p_h$, the linearity of ${\mathit{u}}_h^{⁎}(\cdot )$ along with (8.1) gives

$$\Vert p-{\lambda }_h\Vert \le C[\Vert {\mathit{u}}_h^{⁎}(p)-{\mathit{u}}_h^{⁎}({\lambda }_h)\Vert \le C[\Vert {\mathit{u}}_h(p)-{\mathit{u}}_h({\lambda }_h)\Vert ]\le C[\Vert {\mathit{u}}_h(p)-\mathit{u}\Vert +\Vert \mathit{u}-{\mathit{u}}_h\Vert ].$$ (8.4)

The standard mixed method for (4.4) – (4.6) and (8.2) – (8.3) yields [40]

$${\Vert {\mathit{u}}_h(p)-\mathit{u}\Vert }_{K_i}\le C{h}^r{\Vert p\Vert }_{r+1,K_i},1\le r\le k+1.$$ (8.5)

Finally, (6.12), (8.4) and (8.5) conclude that

$${\Vert p-{\lambda }_h\Vert }_{D_h}\le C[h^r{\Vert p\Vert }_{r+1}+\Vert \mathit{u}-{\mathit{u}}_h\Vert ].$$

 □

9. Implementation and numerical results

Before presenting the numerical results, we will briefly describe the implementation of our discrete scheme for the equations under consideration, that is, nonlinear elliptic problems. We will test both the Fixed-Point Iteration (FPI) and Newton-Raphson method (NRM) for the linearization of discrete system. Moreover, we will present the algorithms for both the methods.

9.1. Fixed-point iteration (FPI)

Let $\{{\mathit{u}}_h^0,p_h^0,{\lambda }_h^0\}$ be the initial guess to the solution of discrete nonlinear system (4.4) – (4.6). Then, we construct the sequence $\{{\mathit{u}}_h^n,p_h^n,{\lambda }_h^n\}\in {\mathit{U}}_h\times S_h\times {\mathcal{L}}_h$ satisfying the relation

$$\begin{gathered} {(\alpha (p_h^n){\mathit{u}}_h^{n+1},\mathit{v})}_{K_i}={(p_h^{n+1},\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle {\lambda }_h^{n+1},\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i} \\ -{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus \gamma },\mathit{v}\in {\mathit{U}}_h^i, \end{gathered}$$ (9.1)

$${(\mathrm{\nabla }\cdot {\mathit{u}}_h^{n+1},w)}_{K_i}={(f(p_h^n),w)}_{K_i},w\in S_h^i,$$ (9.2)

$$\sum _{i=1}^{nb}{\langle {\mathit{u}}_h^{n+1}\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i},m\in {\mathcal{L}}_h.$$ (9.3)

Observe that the system (9.1) – (9.3) is linear for each n. Let $D_i={(\alpha (p_h^n){\mathit{u}}_h^{n+1},\mathit{v})}_{K_i},E_i={(p_h^{n+1},\mathrm{\nabla }\cdot \mathit{v})}_{K_i},{\tilde{g}}_i={\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus \gamma },{\tilde{f}}_i={(f(p_h^n),w)}_{K_i}$, then the standard mixed system on each block $K_i$ yields

$$\left[\begin{array}{cc}\hfill D_i\hfill & \hfill -E_i\hfill \\ \hfill -E_i^{\mathsf{T}}\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\alpha }_i\hfill \\ \hfill {\beta }_i\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\tilde{g}}_i\hfill \\ \hfill {\tilde{f}}_i\hfill \end{array}\right],$$ (9.4)

where ${\alpha }_i,{\beta }_i$ denote the degree of freedom of subdomain unknowns ${\mathit{u}}_h$ and $p_h$ respectively for $1\le i\le nb$. Let ${\xi }_i=({\alpha }_i,{\beta }_i),1\le i\le nb$ be the vector related to subdomain unknowns and let μ be the vector associated with mortar degree of freedom. The system (9.1) – (9.3) returns the global matrix form

$$\left[\begin{array}{ccccc}\hfill A_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \cdots \hfill & \hfill B_1\hfill \\ \hfill 0\hfill & \hfill A_2\hfill & \hfill 0\hfill & \hfill \cdots \cdots \hfill & \hfill B_2\hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill A_{nb}\hfill & \hfill B_{nb}\hfill \\ \hfill C_1\hfill & \hfill C_2\hfill & \hfill \cdots \hfill & \hfill C_{nb}\hfill & \hfill 0\hfill & \hfill \hfill \end{array}\right]\left[\begin{array}{c}\hfill {\xi }_1\hfill \\ \hfill {\xi }_2\hfill \\ \hfill ⋮\hfill \\ \hfill {\xi }_{nb}\hfill \\ \hfill \mu \hfill \end{array}\right]=\left[\begin{array}{c}\hfill F_1\hfill \\ \hfill F_2\hfill \\ \hfill ⋮\hfill \\ \hfill F_{nb}\hfill \\ \hfill 0\hfill \end{array}\right],$$ (9.5)

with $B_i={\langle {\lambda }_h^{n+1},\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},C_i=B_i^{\mathsf{T}},F_i=[\tilde{g},\tilde{f}]$ and $A_i$ are the matrices given in (9.4). We mention that each row of the matrix on the left hand side of (9.5) arises from mixed approximation for $1\le i\le nb$. The matrices $B_i$ are corresponding to the local approximation on each ${\gamma }_i$ for $1\le i\le nb$. The last row, i.e., $C_i,1\le i\le nb$ impose the weak continuity of flux variable across the interface ${\gamma }_{ij}$.

The system (9.5) can be reduced to a lower dimensional mortar interface problem involving only the mortar pressure as follows;

$${\xi }_i=A_i^{-1}(F_i-B_i\mu ),1\le i\le nb,$$ (9.6)

$$\sum _{i=1}^{nb}{C}_i{\xi }_i=0,$$ (9.7)

Using (9.6) in (9.7), we see that

$$\sum _{i=1}^{nb}{C}_i(A_i^{-1}(F_i-B_i\mu ))=0,$$

which implies

$$B\mu =L,$$ (9.8)

where $B={\sum }_{i=1}^{nb}{C}_i{A}_i^{-1}{B}_i$ and $L={\sum }_{i=1}^{nb}{C}_i{A}_i^{-1}{F}_i$. Thus eliminating the subdomain unknowns we obtain the interface problem (9.8). Solving (9.8) for μ and substituting in (9.6), we obtain

$${\xi }_i=A_i^{-1}(F_i-B_i{B}^{-1}L).$$

We describe the implementation procedure of FPI for nonlinear elliptic problems in Algorithm 1.

Algorithm 1.

Fixed-point Iteration (FPI) for mortar mixed method.

9.2. Newton Raphson method (NRM)

We now derive the Newton Raphson method (NRM) for our nonlinear mixed system (4.4) – (4.6). Let $\{{\mathit{u}}_h^0,p_h^0,{\lambda }_h^0\}$ be the given initial guess such that

$${\mathit{u}}_h^{n+1}={\mathit{u}}_h^n+\delta {\mathit{u}}_h,p_h^{n+1}=p_h^n+\delta p_h,{\lambda }_h^{n+1}={\lambda }_h^n+\delta {\lambda }_h.$$ (9.9)

for $n\ge 0$ where $\delta {\mathit{u}}_h,\delta p_h,\delta {\lambda }_h$ are the correction terms. Then the system (4.4) – (4.6) implies

$${(\alpha (p_h^{n+1}){\mathit{u}}_h^{n+1},\mathit{v})}_{K_i}={(p_h^{n+1},\mathrm{\nabla }\cdot \mathit{v})}_{K_i}-{\langle {\lambda }_h^{n+1},\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus {\gamma }_i},\mathit{v}\in {\mathit{U}}_h^i,$$ (9.10)

$${(\mathrm{\nabla }\cdot {\mathit{u}}_h^{n+1},w)}_{K_i}={(f(p_h^{n+1}),w)}_{K_i},w\in S_h^i,$$ (9.11)

$$\sum _{i=1}^{nb}{\langle {\mathit{u}}_h^{n+1}\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}=0,m\in {\mathcal{L}}_h.$$ (9.12)

Using (9.9) and Taylor expansion by neglecting the second order and higher terms, the system (9.10) - (9.12) implies

$$\begin{gathered} {(\alpha (p_h^n)\delta {\mathit{u}}_h,\mathit{v})}_{K_i}+{({\alpha }_p^{\prime }(p_h^n){\mathit{u}}_h^n\delta p_h,\mathit{v})}_{K_i}-{(\delta p_h,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}+{\langle \delta {\lambda }_h,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i} \\ =-{(\alpha (p_h^n){\mathit{u}}_h^n,\mathit{v})}_{K_i}+{(p_h^n,\mathrm{\nabla }\cdot \mathit{v})}_{K_i} \\ -{\langle {\lambda }_h^n,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}-{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},\mathit{v}\in {\mathit{U}}_h^i, \end{gathered}$$ (9.13)

$${(\mathrm{\nabla }\cdot \delta {\mathit{u}}_h,w)}_{K_i}-{(f_p^{\prime }(p_h^n)\delta p_h,w)}_{K_i}={(f(p_h^n),w)}_{K_i}-{(\mathrm{\nabla }\cdot {\mathit{u}}_h^n,w)}_{K_i},w\in S_h^i,$$ (9.14)

$$\sum _{i=1}^{nb}{\langle \delta {\mathit{u}}_h\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}=-\sum _{i=1}^{nb}{\langle {\mathit{u}}_h^n\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i},m\in {\mathcal{L}}_h.$$ (9.15)

We mention that (9.13) – (9.15) is a linear algebraic system corresponding to mortar mixed formulation of nonlinear elliptic equations. The system (9.13) – (9.15) can be solved for $\{\delta {\mathit{u}}_h,\delta p_h\}$ for $1\le i\le nb$. Thus for given initial guess $\{{\mathit{u}}_h^0,p_h^0,{\lambda }_h^0\}$ solve (9.13) – (9.15) and update solution by using (9.9) on each step of Newton-Raphson iteration.

Let $\{\delta {\mathit{u}}_h^i,\delta p_h^i,\delta {\lambda }_h^i\}$ denote the correction terms locally on each subdomain $K_i$. Moreover, let ${\alpha }^i,{\gamma }^i,1\le i\le nb$ denote the degree of freedom related to $\delta {\mathit{u}}_h^i$ and $\delta p_h^i$. Let $\delta {\xi }_1=({\alpha }^1,{\gamma }^1),\delta {\xi }_2=({\alpha }^2,{\gamma }^2),\cdots \cdots \delta {\xi }_n=({\alpha }^{nb},{\gamma }^{nb})$. Let $\mathcal{Z}$ be the vector corresponding to degree of freedom of interface unknown $\delta {\lambda }_h$. Then the linear system (9.13) – (9.15) takes the form

$$\left[\begin{array}{cccccc}\hfill A_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \cdots \hfill & \hfill 0\hfill & \hfill B_1\hfill \\ \hfill 0\hfill & \hfill A_2\hfill & \hfill 0\hfill & \hfill \cdots \cdots \hfill & \hfill 0\hfill & \hfill B_2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill A_{nb}\hfill & \hfill B_{nb}\hfill \\ \hfill C_1\hfill & \hfill C_2\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill C_{nb}\hfill & \hfill 0\hfill & \hfill \hfill \end{array}\right]\left[\begin{array}{c}\hfill \delta {\xi }_1\hfill \\ \hfill \delta {\xi }_2\hfill \\ \hfill ⋮\hfill \\ \hfill \delta {\xi }_{nb}\hfill \\ \hfill \mathcal{Z}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill F_1\hfill \\ \hfill F_2\hfill \\ \hfill ⋮\hfill \\ \hfill F_{nb}\hfill \\ \hfill G\hfill \end{array}\right],$$ (9.16)

with

$$\left[\begin{array}{cc}\hfill D_i\hfill & \hfill -E_i+{\tilde{E}}_i\hfill \\ \hfill -E_i^{\mathsf{T}}\hfill & \hfill H_i\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\alpha }_i\hfill \\ \hfill {\beta }_i\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\tilde{g}}_i\hfill \\ \hfill {\tilde{f}}_i\hfill \end{array}\right],$$

representing the local mixed finite element system on each subdomain block $K_i$. Furthermore, $B_i={\langle \delta {\lambda }_h,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i},C_i=B_i^{\mathsf{T}},G=-{\sum }_{i=1}^{nb}{\langle {\mathit{u}}_h^n\cdot {\mathit{\nu }}_i,m\rangle }_{{\gamma }_i}$ and

$$F_i=\left[\begin{array}{c}\hfill -{\langle g,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{\partial K_i\setminus {\gamma }_i}-{(\alpha (p_h^n){\mathit{u}}_h^n,\mathit{v})}_{K_i}-{\langle {\lambda }_h^n,\mathit{v}\cdot {\mathit{\nu }}_i\rangle }_{{\gamma }_i}+{(p_h^n,\mathrm{\nabla }\cdot \mathit{v})}_{K_i}\hfill \\ \hfill {(f(p_h^n),w)}_{K_i}-{(\mathrm{\nabla }\cdot {\mathit{u}}_h^n,w)}_{K_i}\hfill \end{array}\right].$$

Here matrices $A_i$ come from the application of mixed method locally on each subdomain and $B_i$ is the interface matrix associated to interface ${\gamma }_{ij}$. The matrices $C_i$ are corresponding to the condition of weak continuity across the subdomain interface. System (9.16) implies

$$\delta {\xi }_i=A_i^{-1}(F_i-B_i\mathcal{Z}),1\le i\le nb,$$ (9.17)

$$\sum _{i=1}^{nb}{C}_i\delta {\xi }_i=G,$$ (9.18)

Then similar to FPI (as we did to ge (9.8)), we have

$$B\mathcal{Z}=L,$$ (9.19)

with

$$B=\sum _{i=1}^{nb}{C}_i{A}_i^{-1}{B}_i,L=G+\sum _{i=1}^{nb}{C}_i{A}_i^{-1}{F}_i$$

which gives the interface problem containing only mortar pressure variable. Solving (9.19) for $\mathcal{Z}$ and substituting in (9.17) – (9.18), we recover the local subdomain unknowns ${\mathit{u}}_h$ and $p_h$. The implementation procedure for NRM for nonlinear elliptic equation is given in Algorithm 2.

Algorithm 2.

Newton-Raphson method (NRM) for mortar mixed method.

9.3. Numerical results

To confirm the theoretical finding established in the article, we now present computational results. The computational experiments are performed on two dimensional unit square domain with Dirichlet boundary conditions. In all the examples we use tolerance ${10}^{-8}$ to control nonlinear solves. The Discrete errors are calculated in $L^2$ norms. We implement the linearized discrete system (9.1) – (9.3) for FPI and (9.13) – (9.15) for NRM. The implementation procedure for both methods is provided in Algorithm 1, Algorithm 2. We consider four, eight and sixteen subdomains to perform the experiments. In the case of four subdomains our interface is along $x=1/2,y=1/2$. In eight subdomains, the interface is along $x=1/4,1/2,3/4,y=1/2$. For the sixteen subdomains case, we divide the domain so that the interface is along $x=1/4,1/2,3/4,y=1/4,1/2,3/4$. The graphical visualization of domain decomposition is depicted in Figure 1, Figure 2, Figure 3 where interface skeletons are shown by red lines.

Figure 1.

Domain decomposition for nb = 4.

Figure 2.

Domain decomposition for nb = 8.

Figure 3.

Domain decomposition for nb = 16.

The local problem posed on each subdomain is solved by using the lowest order Raviart Thomas space. For the triangle $T\in {\mathcal{T}}_h(R{T}_0)$., the Raviart Thomas space of index k ($R{T}_k$) is defined by

$$R{T}_k(T)={(P_k(T))}^d+\mathit{x}{P}_k(T),k\ge 0,$$

where $P_k(T)$ denote the space of polynomials of degree less than or equal to k on T. Then the Raviart Thomas ($R{T}_k$) approximation space is defined by

$$R{T}_k(T)=\{\mathit{\phi }|\mathit{\phi }\in R{T}_k(T)\mathrm{\text{for all}}T\in {\mathcal{T}}_h^i,\mathit{\phi }\cdot n_{ij}\mathrm{\text{is constant for all}}{e}_{ij}\in {\epsilon }_h\},$$

where ${\epsilon }_h$ is the collection of interior edges of triangulation ${\mathcal{T}}_h^i$ and $n_{ij}$ denote the unit normal to $e_{ij}$. For $k=0$, the lowest order Raviart Thomas space is

$$R{T}_0(T)=\{\mathit{\phi }:T⟶{\mathit{R}}^2,\mathit{\phi }(x)=(a{x}_1+b_1,a{x}_2+b_2),a,b_1,b_2\in \mathit{R}\}$$

The most common choice in the implementation of mixed finite element method is $R{T}_0-P_0$, that is, lowest order Raviart Thomas space for velocity and constant space for pressure, thus we choose

$${\mathit{U}}_h^i=R{T}_0(T)={(P_0(T))}^d+\mathit{x}{P}_0(T),S_h^i(T)=P_0(T).$$

The interelement continuity ensures that ${\mathit{U}}_h^i\subset {\mathit{U}}_i$ and $S_h^i\subset S_i$. The constant in the basis function ${\mathit{\phi }}_i$ is calculated such that

$${\mathit{\phi }}_i\cdot n_j=\{\begin{array}{cc}1,\hfill & \text{if }i=j\hfill \\ 0,\hfill & \text{if}i\ne j.\hfill \end{array}$$

where $n_j$ is the normal vector associated with the sides of triangle. The interface pressure unknown is approximated by the linear polynomial. We considered both the Fixed-point iteration and Newton's Raphson method to solve the nonlinear problem. The FPI and NRM number of iterations are reported which reflect the efficiency of the numerical method.

Example 9.1

We consider the domain Ω a unit square $(0,1)\times (0,1)$ with coefficient $\mathcal{A}(p)=2+p^2$. The exact solution is taken to be

$$p(x,y)=\sin (\pi x)\sin (\pi y).$$ (9.20)

The solution u and right hand side f is calculated using (9.20).

The discrete norm errors (DNEs) calculated in $L^2-\mathrm{\text{errors}}$ and convergence rate (CRs) for Example 9.1 using FPI are presented in Table 9.1. It is observed that the pressure and velocity approximations have optimal order convergence of $\mathcal{O}(h)$. The order of convergence for interface pressure approximation to λ is of $\mathcal{O}(h^2)$. The convergence rates and discrete errors for different level of grids refinements for Example 9.1 for NRM are displayed in 9.2. The table returns optimal order convergence for subdomain scalar and vector variables. For interface pressure approximation, the rate of convergence is $\mathcal{O}(h^2)$. Results of numerical experiments for Example 9.1 on eight subdomains are shown in Table 9.3, Table 9.4. The results are produced using FPI and NRM. We report $L^2$ discrete errors, order of convergence and number of iteration for the iterative methods. We acquired the optimal order convergence of $\mathcal{O}(h^2)$ for both velocity and pressure approximations which matches theoretical results established in previous sections. The interface pressure approximation has convergence of order $\mathcal{O}(h^2)$.

Table 9.1.

DNEs and CRs for Example 9.1 using FPI ; nb = 4.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	9	1.15E+00		1.31E-01		8.16E-03
16	9	5.72E-01	1.01	6.52E-02	1.01	2.19E-03	1.90
32	9	2.86E-01	1.00	3.26E-02	1.00	5.67E-04	1.95
64	9	1.43E-01	1.00	1.63E-02	1.00	1.44E-04	1.98
128	9	7.14E-02	1.00	8.14E-03	1.00	3.63E-05	1.99
256	9	3.57E-02	1.00	4.07E-03	1.00	9.11E-06	1.99
512	9	1.78E-02	1.00	2.03E-03	1.00	2.28E-06	2.00

Table 9.2.

DNEs and CRs for Example 9.1 using NRM; nb = 4.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	4	1.15E+00		1.31E-01		8.16E-03
16	4	5.72E-01	1.01	6.52E-02	1.01	2.19E-03	1.90
32	4	2.86E-01	1.00	3.26E-02	1.00	5.67E-04	1.95
64	4	1.43E-01	1.00	1.63E-02	1.00	1.44E-04	1.98
128	4	7.14E-02	1.00	8.14E-03	1.00	3.63E-05	1.99
256	4	3.57E-02	1.00	4.07E-03	1.00	9.11E-06	1.99
512	4	1.78E-02	1.00	2.03E-03	1.00	2.28E-06	2.00

Table 9.3.

DNEs and CRs for Example 9.1 using FPI; nb = 8.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	10	2.53E+00		2.87E-01		3.72E-02
16	9	1.25E+00	1.02	1.43E-01	1.01	9.34E-03	1.99
32	9	6.20E-01	1.01	7.15E-02	1.00	2.39E-03	1.97
64	9	3.10E-01	1.00	3.57E-02	1.00	6.04E-04	1.98
128	9	1.55E-01	1.00	1.79E-02	1.00	1.52E-04	1.99
256	9	7.74E-02	1.00	8.93E-03	1.00	3.82E-05	1.99
512	9	3.87E-02	1.00	4.47E-03	1.00	9.56E-06	2.00

Table 9.4.

DNEs and CRs for Example 9.1 using NRM; nb = 8.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	5	2.52E+00		2.89E-01		3.80E-02
16	5	1.25E+00	1.01	1.43E-01	1.02	9.51E-03	2.00
32	4	6.20E-01	1.01	7.15E-02	1.00	2.41E-03	1.98
64	4	3.10E-01	1.00	3.57E-02	1.00	6.07E-04	1.99
128	4	1.55E-01	1.00	1.79E-02	1.00	1.52E-04	2.00
256	4	7.74E-02	1.00	8.93E-03	1.00	3.82E-05	1.99
512	4	3.87E-02	1.00	4.47E-03	1.00	9.56E-06	2.00

Numerical results on eight subdomains for Example 9.1 are presented in Table 9.5, Table 9.6. We observed the optimal order convergence for both pressure and velocity approximation while the order the order of convergence for interface pressure is of order $\mathcal{O}(h^2)$.

Table 9.5.

DNEs and CRs for Example 9.1 using FPI; nb = 16.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	9	2.17E+00		2.56E-01		1.57E-02
16	9	1.08E+00	1.01	1.27E-01	1.01	4.20E-03	1.90
32	9	5.38E-01	1.01	6.36E-02	1.00	1.08E-03	1.96
64	9	2.69E-01	1.00	3.18E-02	1.00	2.75E-04	1.97
128	9	1.34E-01	1.01	1.59E-02	1.00	6.93E-05	1.99
256	9	6.72E-02	1.00	7.94E-03	1.00	1.74E-05	1.99
512	9	3.36E-02	1.00	3.97E-03	1.00	4.35E-06	2.00

Table 9.6.

DNEs and CRs for Example 9.1 using NRM; nb = 16.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	4	2.17E+00		2.56E-01		1.68E-02
16	4	1.08E+00	1.01	1.27E-01	1.01	4.33E-03	1.96
32	4	5.38E-01	1.01	6.36E-02	1.00	1.10E-03	1.98
64	4	2.69E-01	1.00	3.18E-02	1.00	2.77E-04	1.99
128	4	1.34E-01	1.01	1.59E-02	1.00	6.95E-05	1.99
256	4	6.72E-02	1.00	7.94E-03	1.00	1.74E-05	2.00
512	4	3.36E-02	1.00	3.97E-03	1.00	4.36E-06	2.00

The graphical comparison of the approximate solution $p_h$, exact solution p, the computed vector field ${\mathit{u}}_h$ and exact vector field u for Example 9.1 is shown in Figure 4, Figure 5. Computed and exact pressure shown in Figs. 4A and 4B respectively. The velocity solutions are shown in Fig. 5A and 5B. We also depict the graphical results for interface pressure solutions and absolute error $|\lambda -{\lambda }_h|$ in the Fig. 6.

Figure 4.

Exact and computed pressure for Example 9.1.

Figure 5.

Exact and computed vector field for Example 9.1.

Figure 6.

The exact solution λ, approximate solution λ_h and absolute error |λ − λ_h| on interface for Example 9.1.

Example 9.2

We consider the domain Ω a unit square $(0,1)\times (0,1)$ with coefficient $\mathcal{A}(p)=2+p^2$. The exact solution is taken to be

$$p(x,y)=xy(1-x)(1-y).$$ (9.21)

The solution u and right hand side f can be calculated using (9.21).

The discrete $L^2-\mathrm{\text{error}}$ and order of convergence for Example (9.21) using FPI on four, eight and sixteen subdomain blocks are given in Table 9.7, Table 9.9, Table 9.11 respectively. The results for the NRM are given in Table 9.8, Table 9.10, Table 9.12.

Table 9.7.

DNEs and CRs for Example 9.2 using FPI; nb = 4.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	4	7.53E-02		8.94E-03		2.21E-03
16	4	3.74E-02	1.03	4.40E-03	1.06	5.65E-04	1.87
32	4	1.86E-02	1.01	2.19E-03	1.02	1.42E-04	1.97
64	4	9.32E-03	1.01	1.09E-03	1.01	3.54E-05	1.99
128	4	4.66E-03	1.00	5.47E-04	1.01	8.85E-06	2.00
256	4	2.33E-03	1.00	2.73E-04	0.99	2.21E-06	2.00
512	4	1.16E-03	1.00	1.37E-04	1.00	5.53E-07	2.00

Table 9.9.

DNEs and CRs for Example 9.2 using FPI; nb = 8.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	4	1.58E-01		2.03E-02		5.57E-03
16	4	7.85E-02	1.01	9.79E-03	1.05	1.48E-03	1.91
32	4	3.92E-02	1.00	4.85E-03	1.01	3.74E-04	1.98
64	4	1.96E-02	1.00	2.42E-03	1.00	9.37E-05	2.00
128	4	9.78E-03	1.00	1.21E-03	1.00	2.34E-05	2.00
256	4	4.89E-03	1.00	6.04E-04	1.00	5.85E-06	2.00
512	4	2.45E-03	1.00	3.02E-04	1.00	1.46E-06	2.00

Table 9.11.

DNEs and CRs for Example 9.2 using FPI; nb = 16.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	4	1.31E-01		1.74E-02		3.47E-03
16	4	6.51E-02	1.01	8.53E-03	1.03	8.53E-04	2.02
32	4	3.25E-02	1.00	4.24E-03	1.01	2.10E-04	2.02
64	4	1.62E-02	1.00	2.12E-03	1.00	5.18E-05	2.02
128	4	8.12E-03	1.00	1.06E-03	1.00	1.29E-05	2.01
256	4	4.06E-03	1.00	5.29E-04	1.00	3.21E-06	2.01
512	4	2.03E-03	1.00	2.65E-04	1.00	8.00E-07	2.00

Table 9.8.

DNEs and CRs for Example 9.2 using NRM; nb = 4.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	3	7.53E-02		8.94E-03		2.21E-03
16	3	3.74E-02	1.01	4.40E-03	1.02	5.65E-04	1.97
32	3	1.86E-02	1.01	2.19E-03	1.01	1.42E-04	1.99
64	3	9.32E-03	1.00	1.09E-03	1.01	3.54E-05	2.00
128	3	4.66E-03	1.00	5.47E-04	0.99	8.85E-06	2.00
256	3	2.33E-03	1.00	2.73E-04	1.00	2.21E-06	2.00
512	3	1.16E-03	1.01	1.37E-04	0.99	5.53E-07	2.00

Table 9.10.

DNEs and CRs for Example 9.2 using NRM ; nb = 8.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	3	1.58E-01		2.03E-02		5.58E-03
16	3	7.85E-02	1.01	9.79E-03	1.05	1.48E-03	1.91
32	3	3.92E-02	1.00	4.85E-03	1.01	3.74E-04	1.98
64	3	1.96E-02	1.00	2.42E-03	1.00	9.37E-05	2.00
128	3	9.78E-03	1.00	1.21E-03	1.00	2.34E-05	2.00
256	3	4.89E-03	1.00	6.04E-04	1.00	5.85E-06	2.00
512	3	2.45E-03	1.00	3.02E-04	1.00	1.46E-06	2.00

Table 9.12.

DNEs and CRs for Example 9.2 using NRM ; nb = 16.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8	3	1.31E-01		1.74E-02		3.47E-03
16	3	6.51E-02	1.01	8.53E-03	1.03	8.53E-04	2.02
32	3	3.25E-02	1.00	4.24E-03	1.01	2.10E-04	2.02
64	3	1.62E-02	1.00	2.12E-03	1.00	5.18E-05	2.02
128	3	8.12E-03	1.00	1.06E-03	1.00	1.29E-05	2.01
256	3	4.06E-03	1.00	5.29E-04	1.00	3.21E-06	2.01
512	3	2.03E-03	1.00	2.65E-04	1.00	8.00E-07	2.00

The number of iteration to converge the FPI and NRM on each level of mesh refinement are also reported. The optimal order convergence rate $\mathcal{O}(h)$ is observed for both velocity and pressure approximation. The interface pressure approximation has convergence of order $\mathcal{O}(h^2)$.

It is observed that the convergence rates for subdomain variables are of order $\mathcal{O}(h)$. The convergence rate for interface pressure is $\mathcal{O}(h^2)$. The graphical comparison of the solutions on subdomains and interface are given in Figure 7, Figure 8. In Fig. 7A and 7B we depicted the exact and approximate pressure while Fig. 8A and 8B present the comparison of exact and computed velocity fields. The exact and approximate vector field for Example 9.1 are depicted in Fig. 5 The graphical results for Example 9.1 for interface pressure are shown in Fig. 9. We showed the exact solution λ, the approximate solution ${\lambda }_h$ and the absolute error $|\lambda -{\lambda }_h|$. To illustrate the effectiveness and flexibility of the method, we present an example with discontinuous coefficients. In Example 9.3, we divide the computational domain into four regions and take $a(p)=2+p$ in lower left and top right regions while $a(p)=2+p^2$ in upper left and lower right regions.

Example 9.3

We consider the domain Ω a unit square $(0,1)\times (0,1)$ with discontinuous coefficient. The exact solution is taken to be

$$p(x,y)=\cos (2\pi x)\cos (2\pi y).$$ (9.22)

The coefficient profile on different region are shown in Fig. 10. We depicted both the three dimensional (Fig. 10A) and two dimensional view (Fig. 10B). This example demonstrates the flexibility and feature of local grids. The result multiblock mixed approach are comparable to those classical mixed method and mixed methods on locally refined grids for matching grids, however, multiblock approach offers great flexibility which is necessary to handle complicated geometries. Moreover, multiblock approach provides parallel computation which improve the efficiency of the method. It is worth mentioning that the multiblock mixed method gives better accuracy when non-matching grids are used.

Figure 7.

Exact and computed pressure for Example 9.2.

Figure 8.

Exact and computed vector field for Example 9.2.

Figure 9.

The exact solution λ, approximate solution λ_h and absolute error |λ − λ_h| on interface for Example 9.2.

Figure 10.

Coefficient profile for Example 9.3.

We obtained the optimal order convergence for pressure, velocity and interface approximations. The graphical comparison of solutions for Example 9.3 is shown in Figure 11, Figure 12. Figs. 11A and 11B compare the exact and computed pressure. The $x-\mathrm{\text{components}}$ exact and computed velocity are displayed in Figs. 12A and 12A and $y-\mathrm{\text{components}}$ of exact and computed velocity are shown in Fig. 12C and 12D respectively. Finally, the exact interface pressure, computed interface pressure and absolute error are depicted in Fig. 13 (top to bottom). The numerical results for Example 9.3 using NRM are similar to that of FPI as shown in Table 9.13 (except the number of iterations) and are not displayed here. The interface exact and computed solutions along with absolute error are depicted in Fig. 12 Comparing the accuracy of numerical results on four, eight and sixteen subdomains. It is evident from Figure 1, Figure 2, Figure 3 that increasing the subdomain block in domain decomposition leads an increase in the interfaces between the blocks. This in turn have an impact on the precision of data transmission between the blocks across the interface. Consequently, an increase in subdomain effect the accuracy of numerical results on subdomain level as witnessed in the above tables. On the other hand, it is important to note that the size of local problem on fewer subdomain block is significantly larger as compare to those on many subdomains. Therefore, scarifying a slight accuracy we can efficiently solve the local problems. Moreover, the order of convergence remains same regardless number of subdomains.

Figure 11.

Exact and computed pressure for Example 9.3.

Figure 12.

Exact and computed flux for Example 9.3.

Figure 13.

The exact solution λ, approximate solution λ_h and absolute error |λ − λ_h| on interface for Example 9.3.

Table 9.13.

DNEs and CRs for Example 9.3 using FPI; nb = 4.

1/h	Iter	‖u − uh‖	CR	‖p − ph‖	CR	‖λ − λh‖	CR
8		4.63E+00		2.69E-01		7.15E-02
16		2.26E+00	1.03	1.32E-01	1.03	1.98E-02	1.85
32		1.12E+00	1.01	6.56E-02	1.01	5.10E-03	1.96
64		5.60E-01	1.00	3.27E-02	1.00	1.29E-03	1.98
128		2.80E-01	1.00	1.64E-02	1.00	3.23E-04	2.00
256		1.40E-01	1.00	8.18E-03	1.00	8.09E-05	2.00
512		7.00E-02	1.00	4.09E-03	1.00	2.02E-05	2.00

Comparing the numerical results obtained by FPI and NRM, the Tables show that there is no difference in accuracy and order of convergence however; NRM converges faster than FPI. Also we obtained same order of convergence for both the iterative methods.

10. Conclusion

We consider the numerical approximation of second order nonlinear elliptic equations modeling Darcy flow in porous medium with pressure dependent permeability. This paper presents the generalization multiblock mortar mixed method for the nonlinear partial differential equations. The existence of the discrete system for our method is established by using the fixed point theorem. We also demonstrate the uniqueness of solution. Optimal error estimates are derived for velocity and pressure approximations and an error bound for the interface pressure is also presented. We briefly describe the implementation procedure for our formulation and present the algorithm to perform computational experiments. We have used both the fixed point iteration and Newton Raphson method for the linearization of nonlinear term. Numerical Examples are presented to verify the theoretical order of convergence.

CRediT authorship contribution statement

Muhammad Arshad: Writing – original draft, Methodology, Conceptualization. Adil Mehmood: Writing – original draft, Software, Investigation. Zabidin Salleh: Resources, Funding acquisition. Sumaira Saleem Akhtar: Visualization, Software, Investigation. Suliman Khan: Validation, Resources. Mustafa Inc: Writing – review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.