A Parallel iterative Algorithm for primal-dual weak Galerkin Schemes

TL;DR

This paper introduces a parallel iterative domain decomposition algorithm for primal-dual weak Galerkin finite element methods, providing theoretical analysis and error estimates for solving the Poisson equation efficiently.

Contribution

It develops a novel parallelizable iterative scheme for PDWG methods with proven convergence and optimal error estimates, enhancing computational efficiency for PDE solutions.

Findings

Proven existence and uniqueness of PDWG solutions.

Derived optimal error estimates in discrete and L2 norms.

Established convergence of the domain decomposition iterative method.

Abstract

This paper presents and analyzes a parallelizable iterative procedure based on domain decomposition for primal-dual weak Galerkin (PDWG) finite element methods applied to the Poisson equation. The existence and uniqueness of the PDWG solution are established. Optimal order of error estimates are derived in both a discrete norm and the $L^2$ norm. The convergence analysis is conducted for domain decompositions into individual elements associated with the PDWG methods, which can be extended to larger subdomains without any difficulty.