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

Haoran Xu, Kunyang Li, Xingye Yue

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.

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

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 accuracy. For Neumann boundaries, a discrete specular reflection mechanism is employed, yielding convergence. Periodic boundaries are handled through modular wrapping, also achieving 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.

Paper Structure

This paper contains 14 sections, 3 theorems, 98 equations, 3 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Numerical Scheme
  3. Numerical Scheme for Boundary Conditions of the First Type
  4. Parabolic Partial Differential Equations
  5. Elliptic Partial Differential Equations
  6. Reflective Boundary Conditions
  7. Periodic Boundary Conditions
  8. Numerical Results
  9. Dirichlet Boundary Conditions
  10. Numerical Results
  11. Discussion
  12. Homogeneous Neumann Boundary Conditions
  13. Periodic Boundary Conditions
  14. Conclusion

Key Result

Theorem 1

Let $f$ be sufficiently smooth and consider the mesh scaling $h = h_1 = h_2$. Define the pointwise error at grid node $(i,j)$ by Then the numerical solution produced by scheme alg:scheme-elliptic-dirichletsatisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Illustration of the Algorithm \ref{['alg:scheme-elliptic-dirichlet']} , where $s_1 = s_2 = s_4 = h$ and $s_3 <h$.
  • Figure 2: Illustration of Algorithm \ref{['alg:neumann-scheme']}. Here, $X_4^h < x_0$ and $Y_3^h < y_0$ are reflected back into the domain.
  • Figure 3: Schematic illustration of Algorithm \ref{['alg:periodic']}, showing periodic wrapping of branch endpoints.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3: Improved Convergence under Periodicity
  • proof : Proof sketch