An $L^p$- Primal-Dual Weak Galerkin method for div-curl Systems

TL;DR

This paper introduces an $L^p$-primal-dual weak Galerkin finite element method for div-curl systems, achieving accurate solutions under low regularity and effectively approximating harmonic fields on complex domains.

Contribution

The paper develops a novel $L^p$-PDWG finite element scheme that handles low regularity and complex topologies, with proven error estimates and numerical validation.

Findings

Provides optimal error estimates in $L^q$-norm.

Demonstrates effectiveness through numerical experiments.

Handles low regularity and complex topologies effectively.

Abstract

This paper presents a new $L^p$-primal-dual weak Galerkin (PDWG) finite element method for the div-curl system with the normal boundary condition for $p>1$. Two crucial features for the proposed $L^p$-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div-curl system under the low $W^{\alpha, p}$-regularity ($\alpha>0$) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the $L^q$-norm for the primal variable where $\frac{1}{p}+\frac{1}{q}=1$. A series of numerical experiments are presented to demonstrate the performance of the proposed $L^p$-PDWG algorithm.