Efficient optimization-based quadrature for variational discretization of nonlocal problems

Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates use of vector calculus to prove well-posedeness, convergence, stability such schemes. Employing an FE also meshing complicated domain geometries coupling methods for local problems. However, weak involve computation a double-integral, which is computationally expensive presents several challenges. In particular, inner integral associated stiffness matrix defined over intersections mesh elements ball radius $\delta$, where $\delta$ range interaction. Identifying parameterizing these nontrivial computational geometry problem. this work, we propose quadrature technique integration performed using points distributed full ball, without regard how it intersects elements, weights are computed based on generalized moving least squares method. Thus, as opposed all previously employed methods, our does not require element-by-element fully circumvents element-ball intersections. This paper considers one- two-dimensional implementations piecewise linear continuous approximations, focusing case size h proportional, typical practical computations. When boundary conditions treated carefully outer accurately, proposed asymptotically compatible limit $h \sim \delta \to 0$, featuring at first-order convergence L^2 dimensions, both uniform nonuniform grids.