A regularized truncated finite element method for degenerate parabolic stochastic PDE on non-compact graph

TL;DR

This paper develops a novel finite element-based numerical method for approximating degenerate parabolic stochastic PDEs on non-compact graphs, addressing challenges like non-compactness and degeneracy.

Contribution

It introduces a multi-step strategy combining graph truncation, regularization, and finite element discretization, with proven strong convergence in weighted spaces.

Findings

Proposed scheme converges strongly in weighted L2-space.

Method effectively handles non-compactness and degeneracy.

Framework extensible to broader classes of graphs and operators.

Abstract

We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise perturbations. The numerical discretization of these equations faces several challenges, including the non-compactness of the graph, the degeneracy of the differential operator near vertices, and the non-symmetry of the associated bilinear form. To address these issues, we propose a multi-step numerical strategy combining graph truncation, localized coefficient regularization, and finite element spatial discretization. By incorporating localization techniques, tightness arguments, and resolvent estimates, we establish the strong convergence of the proposed scheme in a weighted $L^2$-space. Our results provide a systematic methodology that is potentially extensible to more general non-compact graphs and degenerate operators.