An arbitrary order Reconstructed Discontinuous Approximation to Fourth-order Curl Problem

TL;DR

This work develops an arbitrary-order reconstructed discontinuous Galerkin method for the fourth-order curl problem, leveraging a patch-based local least-squares reconstruction to form a low-DOF, high-accuracy space within a symmetric IPDG framework. The authors prove coercivity, continuity, and optimal a priori error estimates in the energy norm and in $L^2$, assuming a uniform bound on the reconstruction constant $\Lambda_m$, and validate the theory with numerical experiments in 2D and 3D. The reconstruction-based approach yields high-order convergence with fewer degrees of freedom, offering a flexible and efficient tool for complex electromagnetism and MHD applications. The work also provides practical guidance for computing $\Lambda_m$ and discusses future directions for efficiency and extensions.

Abstract

We present an arbitrary order discontinuous Galerkin finite element method for solving the fourth-order curl problem using a reconstructed discontinuous approximation method. It is based on an arbitrarily high-order approximation space with one unknown per element in each dimension. The discrete problem is based on the symmetric IPDG method. We prove a priori error estimates under the energy norm and the L^2 norm and show numerical results to verify the theoretical analysis.

Figures (3)

• Figure 1: The convergence histories under the $\| \cdot \|_{L^2(\Omega)}$ (left) and the$\| \cdot\|_{\mathrm{DG}}$ (right) in Example 1.
• Figure 2: The convergence histories under the $\| \cdot \|_{L^2(\Omega)}$ (left) and the $\| \cdot\|_{\mathrm{DG}}$(right) in Example 2.
• Figure 3: $\Lambda_m$ in 2d with $h = 1/40$ (row 1) / $\Lambda_m$ in 3d with $h = 1/16$ (row 2).

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Remark 1
• Lemma 6