Meshfree method for solving the elliptic Monge-Ampère equation with Dirichlet boundary

Abstract

We develop the meshfree method for solving the elliptic Monge-Ampère equation with Dirichlet boundary on the bounded domain and prove its convergence in the paper. In terms of trial, we use the radial functions (for example, Whittle-Matérn-Sobolev kernels and Wendland's compactly supported radial basis functions) which can reproduce $W^{\sigma ,2}\left(R^d\right)$ to construct the finite dimensional approximate spaces. It allows the easy construction of approximation spaces in arbitrary dimensions with arbitrary smoothness and avoids the huge workload caused by mesh-based methods at the same time. In terms of testing, it just needs to take values directly on the collocation points and greatly simplifies the difficulties caused by variation and integration. Theoretically, we obtain $L^2$ error when the testing discretization is finer than the trial discretization. The convergence rate depends on the regularity of the solution, the smoothness of the computing domain, and the approximation of kernel-based trial spaces. An extension to non-Dirichlet boundary condition is in a forthcoming paper.