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

TL;DR

This paper introduces a high-order finite element method using reconstructed spaces for the biharmonic equation, enabling optimal convergence and efficient preconditioning with mesh-independent condition numbers.

Contribution

It develops a novel reconstructed approximation space with high-order accuracy sharing nodal degrees of freedom with linear space, and introduces an optimal preconditioning strategy for the resulting linear system.

Findings

Achieves arbitrarily high-order accuracy in solving biharmonic equations.

Provides a preconditioner with a condition number independent of mesh size.

Demonstrates the method's effectiveness through numerical experiments in 2D and 3D.

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.