Random walk on boundary with iterative refinement for solving interior Dirichlet and exterior Neumann problems for the modified Helmholtz equation

This paper develops and analyzes a hybrid meshfree Random Walk on Boundary (RWB) algorithm with iterative refinement for solving two-dimensional boundary value problems. The method is applied to the interior Dirichlet problem and the exterior Neumann problem for the modified Helmholtz equation. A rigorous proof of the exponential convergence of the proposed iterative refinement procedure is established, together with error and computational complexity estimates. The convergence analysis applies to both convex and non-convex domains, thereby overcoming a major limitation of the classical RWB algorithm. Compared with the standard RWB method, the proposed approach achieves dramatically higher accuracy and significantly improved computational efficiency, with the advantage increasing as higher solution accuracy is required. Numerical experiments confirm the theoretical predictions and demonstrate the effectiveness of the proposed algorithm. The proposed framework extends straightforwardly to three-dimensional boundary value problems for the modified Helmholtz equation.