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

Ruo Li, Qicheng Liu, Fanyi Yang

TL;DR

This work develops a preconditioned interior-penalty discontinuous Galerkin method for elliptic problems using a reconstructed discontinuous approximation that achieves high-order accuracy with only one degree of freedom per element. A constrained local least-squares reconstruction maps piecewise constants to a high-order space $U_h^m$, and a norm equivalence to the piecewise-constant space enables an optimal preconditioner built from $U_h^0$, yielding a condition number bound independent of the mesh size. The method preserves standard $O(h^{-2})$ conditioning of DG discretizations while dramatically reducing DOFs and keeping efficiency via a robust A0-based preconditioner and multigrid/blocking strategies. Numerical experiments in 2D and 3D, including polygonal meshes, validate the optimal convergence rates and the mesh-size–independent performance of the preconditioned system, highlighting practical impact for large-scale high-order DG simulations.

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.

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

TL;DR

This work develops a preconditioned interior-penalty discontinuous Galerkin method for elliptic problems using a reconstructed discontinuous approximation that achieves high-order accuracy with only one degree of freedom per element. A constrained local least-squares reconstruction maps piecewise constants to a high-order space , and a norm equivalence to the piecewise-constant space enables an optimal preconditioner built from , yielding a condition number bound independent of the mesh size. The method preserves standard conditioning of DG discretizations while dramatically reducing DOFs and keeping efficiency via a robust A0-based preconditioner and multigrid/blocking strategies. Numerical experiments in 2D and 3D, including polygonal meshes, validate the optimal convergence rates and the mesh-size–independent performance of the preconditioned system, highlighting practical impact for large-scale high-order DG simulations.

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.
Paper Structure (14 sections, 10 theorems, 83 equations, 4 figures, 16 tables, 1 algorithm)

This paper contains 14 sections, 10 theorems, 83 equations, 4 figures, 16 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Reconstructed Discontinuous Approximation Method
  4. Approximation to the elliptic problem
  5. Numerical Results
  6. Study on convergence rate
  7. Example 1.
  8. Example 2.
  9. Study on preconditioned system
  10. Example 3.
  11. Example 4.
  12. Example 5.
  13. Example 6.
  14. Conclusions

Key Result

Lemma 1

For each $K \in \mathcal{T}_h$, the problem eq_leastsquares admits a unique solution.

Figures (4)

  • Figure 1: The triangular mesh (left), polygonal mesh (middle) and tetrahedral mesh (right).
  • Figure 2: The comparison of the $L^2$ error for the spaces $U_h^m$ and $V_h^m$ in Example 1 (left) / Example 2 (right).
  • Figure 3: The convergence histories of CG (left) /GMRES (right) solvers on the mesh $h = 1/160$ (solid line: CG/GMRES solver with preconditioner $A_0^{-1}$, dashed line: standard CG/GMRES solver).
  • Figure 4: The convergence histories of CG (left) /GMRES (right) solvers on the mesh $h = 1/32$ (solid line: CG/GMRES solver with preconditioner $A_0^{-1}$, dashed line: standard CG/GMRES solver).

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • Remark 2
  • Lemma 5
  • ...and 8 more