A Preconditioned Discontinuous Galerkin Method for Biharmonic Equation with $C^0$-Reconstructed Approximation

TL;DR

This work introduces a preconditioned discontinuous Galerkin method for the biharmonic equation by constructing a high-order space $U_h^m$ via local patch-based least-squares reconstruction from the $C^0$ linear space. The reconstructed space shares the same nodal DOFs as the linear space, enabling a straightforward interior penalty DG discretization with optimal convergence rates, while establishing a norm equivalence to enable an efficient, mesh-size–robust preconditioner derived from the lowest-order space. A norm-equivalence-based preconditioner $A_L^{-1}$ yields a condition number for $A_L^{-1} A_m$ that is independent of the mesh size, and a multigrid strategy is proposed to approximate $A_L^{-1}$. Numerical results in 2D and 3D validate high-order accuracy and the efficiency of the preconditioner, showing near-constant iteration counts under mesh refinement and reduced DOFs for comparable accuracy compared to traditional IPDG methods.

Abstract

In this paper, we present a high-order finite element method based on a reconstructed approximation to the biharmonic equation. In our construction, the space is reconstructed from nodal values by solving a local least squares fitting problem per element. It is shown that the space can achieve an arbitrarily high-order accuracy and share the same nodal degrees of freedom with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme can be directly applied to the reconstructed space for solving the biharmonic equation. We prove that the numerical solution converges with optimal orders under error measurements. More importantly, we establish a norm equivalence between the reconstructed space and the continuous linear space. This property allows us to precondition the linear system arising from the high-order space by the linear space on the same mesh. This preconditioner is shown to be optimal in the sense that the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size. Numerical examples in two and three dimensions are provided to illustrate the accuracy of the scheme and the efficiency of the preconditioning method.

Figures (6)

• Figure 1: The triangular mesh (left) and tetrahedral mesh (right).
• Figure 2: The numerical errors under the energy norm (left)/$L^2$ norm (right) in Example 1.
• Figure 3: The numerical comparison of $L^2$ errors for different methods (from left to right, $m = 2, 3, 4$).
• Figure 4: The numerical errors under the energy norm (left)/$L^2$ norm (right) in Example 2.
• Figure 5: The constant $\Lambda_m$ in two dimensions.

Theorems & Definitions (29)

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