A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions

TL;DR

This paper introduces a primal-dual weak Galerkin finite element method for div-curl systems that performs accurately under low regularity conditions and effectively approximates normal harmonic vector fields, demonstrated through numerical experiments.

Contribution

It develops a novel PDWG finite element scheme that handles low regularity div-curl problems and approximates harmonic fields regardless of domain topology.

Findings

Accurate solutions under low regularity assumptions.

Effective approximation of normal harmonic vector fields.

Validated through seven numerical experiments.

Abstract

This paper presents a numerical method for div-curl systems with normal boundary conditions by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low $H^\alpha$-regularity ($\alpha>0$) assumption for the true solution, but also an effective approximation of normal harmonic vector fields regardless the topology of the domain. Results of seven numerical experiments are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields.