Stability of a generalized particle method for a Poisson equation by discrete Sobolev norms

TL;DR

This paper analyzes the stability of a generalized particle method for solving the Poisson equation, introducing discrete Sobolev norms and connectivity conditions to establish unique solvability and stability of the discretization.

Contribution

The paper introduces a novel stability analysis framework for a generalized particle method using discrete Sobolev norms and connectivity conditions.

Findings

Unique solvability of the discretized Poisson equation is established.

Stability is proven under semi-regularity of discrete parameters.

Connectivity condition ensures well-posedness of the particle discretization.

Abstract

Numerical analysis is conducted for a generalized particle method for a Poisson equation. Unique solvability is derived for the discretized Poisson equation by introducing a connectivity condition for particle distributions. Moreover, by introducing discrete Sobolev norms and a semi-regularity of a family of discrete parameters, stability is obtained for the discretized Poisson equation based on the norms.