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A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities

Xiajie Huang, Fei Wang, Weimin Han, Min Ling

TL;DR

The work addresses solving $H(\mathrm{curl})$-elliptic hemivariational inequalities arising from Maxwell-type problems with nonmonotone constitutive laws. It introduces an Interior Penalty Discontinuous Galerkin (IPDG) discretization equipped with appropriate numerical fluxes to stabilize the scheme and handle the nonmonotone Clarke subdifferential term $\psi^0$. The authors establish consistency, boundedness, and stability of the IPDG formulation, prove existence and uniqueness of the discrete solution via a strongly convex energy functional, and derive a priori error estimates that yield optimal convergence under regularity assumptions; linear elements achieve first-order convergence in the energy norm under mild regularity. A numerical example corroborates the theoretical rates, demonstrating the method’s effectiveness and potential for solving HVIs in electromagnetics with nonmonotone constitutive behavior.

Abstract

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities

TL;DR

The work addresses solving -elliptic hemivariational inequalities arising from Maxwell-type problems with nonmonotone constitutive laws. It introduces an Interior Penalty Discontinuous Galerkin (IPDG) discretization equipped with appropriate numerical fluxes to stabilize the scheme and handle the nonmonotone Clarke subdifferential term . The authors establish consistency, boundedness, and stability of the IPDG formulation, prove existence and uniqueness of the discrete solution via a strongly convex energy functional, and derive a priori error estimates that yield optimal convergence under regularity assumptions; linear elements achieve first-order convergence in the energy norm under mild regularity. A numerical example corroborates the theoretical rates, demonstrating the method’s effectiveness and potential for solving HVIs in electromagnetics with nonmonotone constitutive behavior.

Abstract

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

Paper Structure

This paper contains 8 sections, 12 theorems, 131 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The DG method for H(curl)-elliptic HVI
  4. Error analysis
  5. Properties of the IPDG scheme
  6. Existence and uniqueness
  7. A priori error estimate
  8. Numerical example

Key Result

Lemma 1

Let $V$ be a real Banach space, and let $g\colon V \to \mathbb{R}$ be locally Lipschitz continuous. Then $g$ is strongly convex on $V$ with a constant $\alpha>0$ if and only if $\partial g$ is strongly monotone on $V$ with a constant $2\alpha$, i.e.,

Figures (3)

  • Figure 1: Two adjacent tetrahedral elements and a common face
  • Figure 2: Convergence of the numerical solutions
  • Figure 3: Streamlines of numerical solutions

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1: fan2003generalized
  • Proposition 1: han2020minimization
  • Theorem 1: Consistency
  • proof
  • Lemma 2: Boundedness
  • proof
  • Lemma 3: Stability
  • ...and 13 more