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A Finite Element Method by Patch Reconstruction for the Quad-Curl Problem Using Mixed Formulations

Ruo Li, Qicheng Liu, Shuhai Zhao

TL;DR

This work tackles the quad-curl problem by designing a mixed discontinuous Galerkin method that leverages a divergence-free auxiliary variable and a patch-reconstructed space to achieve high-order accuracy with a minimal increase in degrees of freedom. By applying penalty terms, the method avoids discrete inf-sup constraints while attaining optimal convergence in the energy norm and suboptimal convergence in the $L^2$ norm, with a duality-based argument used for the latter. The theoretical results are supported by numerical experiments in both two and three dimensions, confirming the expected rates and demonstrating practical efficiency. The approach offers a flexible, high-order, low-DOF framework suitable for complex electromagnetic-related quad-curl problems on general meshes.

Abstract

We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variables is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal $L^2$ convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.

A Finite Element Method by Patch Reconstruction for the Quad-Curl Problem Using Mixed Formulations

TL;DR

This work tackles the quad-curl problem by designing a mixed discontinuous Galerkin method that leverages a divergence-free auxiliary variable and a patch-reconstructed space to achieve high-order accuracy with a minimal increase in degrees of freedom. By applying penalty terms, the method avoids discrete inf-sup constraints while attaining optimal convergence in the energy norm and suboptimal convergence in the norm, with a duality-based argument used for the latter. The theoretical results are supported by numerical experiments in both two and three dimensions, confirming the expected rates and demonstrating practical efficiency. The approach offers a flexible, high-order, low-DOF framework suitable for complex electromagnetic-related quad-curl problems on general meshes.

Abstract

We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variables is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.
Paper Structure (6 sections, 11 theorems, 77 equations, 3 figures, 1 table)

This paper contains 6 sections, 11 theorems, 77 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Reconstructed Discontinuous Space
  4. Approximation to Quad-Curl Problem
  5. Numerical Results
  6. Conclusion

Key Result

Theorem 1

2019cc Given $\bm{f}\in H(\mathrm \text{div}^0,\Omega)$, the problem eq_quadcurl2 admits a unique solution $(\bm{u},p)\in H_0(\mathrm \text{curl}^2,\Omega)\times H_0^1(\Omega)$ with and $p=0$.

Figures (3)

  • Figure 1: 2d triangular partition with $h = 1/10$ (left) / 3d tetrahedral partition with $h = 1/4$ (right).
  • Figure 2: The convergence histories under the $\| \cdot \|_{L^2(\Omega)}$ (left) and the$\| \cdot\|_{\mathrm{DG}}$ (right) in Example 1.
  • Figure 3: The convergence histories under the $\| \cdot \|_{L^2(\Omega)}$ (left) and the $\| \cdot\|_{\mathrm{DG}}$(right) in Example 2.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 7 more