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Finite Volume Analysis of the Poisson Problem via a Reduced Discontinuous Galerkin Space

Wenbo Hu, Yinhua Xia

TL;DR

This work develops a high-order finite-volume method for the Poisson problem by leveraging a reduced discontinuous Galerkin (RDG) space as the trial space and a piecewise-constant test space, forming a Petrov–Galerkin IPDG framework. It proves norm equivalence and stability, delivering optimal convergence in the DG energy norm and suboptimal behavior in the $L^2$ norm, with rigorous analysis supported by 1D and 2D Dirichlet and periodic boundary conditions. The RDG reconstruction crucially lowers degrees of freedom while preserving high-order accuracy, and numerical experiments confirm the theoretical rates and efficiency. By bridging finite-volume and discontinuous-Galerkin methodologies, the paper provides a solid mathematical foundation for high-order, locally conservative FVMs on structured grids, with potential extensions to nonlinear and time-dependent problems.

Abstract

In this paper, we propose and analyze a high-order finite volume method for the Poisson problem based on the reduced discontinuous Galerkin (RDG) space. The main idea is to employ the RDG space as the trial space and the piecewise constant space as the test space, thereby formulating the scheme in a Petrov-Galerkin framework. This approach inherits the local conservation property of finite volume methods while benefiting from the approximation capabilities of discontinuous Galerkin spaces with significantly fewer degrees of freedom. We establish a rigorous error analysis of the proposed scheme: in particular, we prove optimal-order convergence in the DG energy norm and suboptimal-order convergence in \(L^2\) norm. The theoretical analysis is supported by a set of one- and two-dimensional numerical experiments with Dirichlet and periodic boundary conditions, which confirm both the accuracy and efficiency of the method. The significance of this work lies in bridging finite volume and discontinuous Galerkin methodologies through the RDG space, thus enabling finite volume schemes with a mathematically rigorous convergence theory.

Finite Volume Analysis of the Poisson Problem via a Reduced Discontinuous Galerkin Space

TL;DR

This work develops a high-order finite-volume method for the Poisson problem by leveraging a reduced discontinuous Galerkin (RDG) space as the trial space and a piecewise-constant test space, forming a Petrov–Galerkin IPDG framework. It proves norm equivalence and stability, delivering optimal convergence in the DG energy norm and suboptimal behavior in the norm, with rigorous analysis supported by 1D and 2D Dirichlet and periodic boundary conditions. The RDG reconstruction crucially lowers degrees of freedom while preserving high-order accuracy, and numerical experiments confirm the theoretical rates and efficiency. By bridging finite-volume and discontinuous-Galerkin methodologies, the paper provides a solid mathematical foundation for high-order, locally conservative FVMs on structured grids, with potential extensions to nonlinear and time-dependent problems.

Abstract

In this paper, we propose and analyze a high-order finite volume method for the Poisson problem based on the reduced discontinuous Galerkin (RDG) space. The main idea is to employ the RDG space as the trial space and the piecewise constant space as the test space, thereby formulating the scheme in a Petrov-Galerkin framework. This approach inherits the local conservation property of finite volume methods while benefiting from the approximation capabilities of discontinuous Galerkin spaces with significantly fewer degrees of freedom. We establish a rigorous error analysis of the proposed scheme: in particular, we prove optimal-order convergence in the DG energy norm and suboptimal-order convergence in norm. The theoretical analysis is supported by a set of one- and two-dimensional numerical experiments with Dirichlet and periodic boundary conditions, which confirm both the accuracy and efficiency of the method. The significance of this work lies in bridging finite volume and discontinuous Galerkin methodologies through the RDG space, thus enabling finite volume schemes with a mathematically rigorous convergence theory.

Paper Structure

This paper contains 26 sections, 30 theorems, 207 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Function spaces
  4. Meshes and notation
  5. Cell averages and projection operators
  6. Discrete function spaces
  7. Interface operators and norms
  8. Auxiliary inequalities
  9. Reduced Discontinuous Galerkin Space
  10. Reconstruction operator
  11. Properties of the reconstruction operator
  12. Reduced discontinuous Galerkin space
  13. Analysis of the Finite Volume Method for the Poisson Problem
  14. Dirichlet Problem
  15. 1D Error Estimates
  16. ...and 11 more sections

Key Result

Lemma 3.1

If Assumption assumption1 holds, then the reconstruction is $k$-exact: Moreover, if $u\in H^{k+1}(S(K))$, then where $C$ is independent of $h$.

Figures (2)

  • Figure 5.1: Convergence of the errors in the $\| \cdot \|_{L^2(\Omega)}$ and $\| \cdot \|_{{\mathrm{sip}{,}\ast}}$ norms for the Dirichlet problem in one dimension (left) and two dimensions (right).
  • Figure 5.2: Convergence of the errors in the $\| \cdot \|_{L^2(\Omega)}$ and $\| \cdot \|_{{\mathrm{sip}{,}\ast}}$ norms for the periodic problem in one dimension (left) and two dimensions (right).

Theorems & Definitions (54)

  • Lemma 3.1: hou2024rdg
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3: Norm equivalence
  • proof
  • Lemma 4.4: Reconstruction boundedness
  • proof
  • ...and 44 more