Preconditioned NonSymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

TL;DR

This paper introduces an efficient preconditioning approach for elliptic problems using reconstructed discontinuous approximation, enabling high-order polynomial spaces with minimal degrees of freedom, improving computational efficiency and robustness.

Contribution

The paper develops a novel preconditioning method based on reconstructed high-order spaces for DG methods, ensuring mesh-independent condition numbers and enhanced efficiency.

Findings

Condition number bound is mesh-independent.

Reconstructed high-order space is norm-equivalent to piecewise constant space.

Numerical results confirm theoretical efficiency and validity.

Abstract

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. We reconstruct a high-order piecewise polynomial space that arbitrary order can be achieved with one degree of freedom per element. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin method. Compared with the standard DG method, we can enjoy the advantage on the efficiency of the approximation. Besides, we establish an norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further allows us to construct an optimal preconditioner from the piecewise constant space. The upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is shown to be independent of the mesh size. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.