A Non-compact Positivity-Preserving Numerical Scheme for Elliptic Differential Equations Based on Mathematical Expectation

TL;DR

This paper introduces a new non-compact, positivity-preserving numerical scheme for elliptic equations, leveraging probabilistic representations and designed for anisotropic diffusion with robust boundary condition handling.

Contribution

It develops a wide-stencil, expectation-based scheme that maintains positivity and stability, effective for complex anisotropic diffusion problems with various boundary conditions.

Findings

Achieves $O(h)$ accuracy for Dirichlet and periodic boundaries.

Attains $O(h^{1/2})$ convergence for Neumann boundaries.

Numerical experiments confirm theoretical convergence rates.

Abstract

We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form elliptic equations. Based on the Feynman--Kac formula, the solution is represented as a conditional expectation associated with a diffusion process.Instead of using compact Markov chain approximations, we construct a wide-stencil scheme by approximating the expectation with carefully designed transition probabilities, ensuring both consistency and positivity preservation. The method is effective for anisotropic diffusion problems with mixed derivatives, where classical schemes typically fail unless the covariance matrix is diagonally dominant. A key feature of the proposed framework is its robust treatment of boundary conditions. For Dirichlet boundaries, we introduce a quadtree-based non-uniform stopping-time strategy, achieving $O(h)$ accuracy. For Neumann boundaries, a discrete specular reflection mechanism is employed, yielding $O(h^{1/2})$ convergence. Periodic boundaries are handled through modular wrapping, also achieving $O(h)$ accuracy. The resulting schemes are unconditionally stable and positivity-preserving due to their probabilistic structure. Numerical experiments confirm the theoretical convergence rates under all boundary conditions considered.