Table 4.

Trial function library for nonlocal 2D example (Section 5.3). Interaction potentials ${[K]}_{\delta }$ indicate cutoff potentials of the form (5.6) with $\delta =0.01$ such that the resulting potential is Lipschitz.

Mean-field term	Trial function library
$\nabla \cdot (U\nabla K\ast U)$	$\{\begin{array}{cc}\nabla \cdot (U\nabla {\mid x\mid }^m\ast U)\hfill & m\in \{2,3,4,5,6\}\hfill \\ \nabla \cdot (U\nabla {\left[{\mid x\mid }^{1∕2}\right]}_{\delta }\ast U)\hfill & \hfill \\ \nabla \cdot (U\nabla {[\mid x\mid (\log \mid x\mid -1)]}_{\delta }\ast U)\hfill & \hfill \\ \nabla \cdot (U\nabla {[\log \mid x\mid ]}_{\delta }\ast U)\hfill & \hfill \end{array}\phantom{\}}$
$\nabla \cdot (U\nabla V)$	${\partial }_{x_i}\left(U{x}_1^m{x}_2^n\right)0\le m+n\le 5,i\in \{1,2\}$
$\frac{1}{2}{\sum }_{i,j=1}^d\frac{{\partial }^2{(U\sigma {\sigma }^T)}_{ij}}{\partial x_i\partial x_j}$	$\Delta (U\cos (m{x}_1)\cos (n{x}_2))$, $(m,n)\in \{0,1,2\}$