A Parallel iterative Algorithm for primal-dual weak Galerkin Schemes

TL;DR

This work develops a parallelizable domain-decomposition algorithm for the primal-dual weak Galerkin (PDWG) discretization of the Poisson problem $\Delta u=f$ in $\Omega$, with Dirichlet data on $\Gamma$. The authors establish existence and uniqueness of the PDWG solution, derive optimal order error estimates in both discrete and $L^2$ norms, and formulate a subdomain-based iterative scheme that enables scalable parallel computation. They also provide a convergence analysis of the domain-decomposition iterative method, showing that interface residuals decay and the method converges to the PDWG solution. Collectively, the results extend PDWG methods to large-scale, parallel architectures while retaining provable stability and accuracy for elliptic problems.

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.

Theorems & Definitions (18)

• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof