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

TL;DR

This paper introduces a high-order finite volume method for the Poisson problem using a reduced discontinuous Galerkin space, achieving local conservation and optimal convergence with fewer degrees of freedom.

Contribution

It develops a novel Petrov-Galerkin finite volume scheme based on RDG space, combining finite volume conservation with DG approximation properties and providing rigorous error analysis.

Findings

Optimal-order convergence in DG energy norm

Suboptimal convergence in L2 norm

Numerical experiments confirm accuracy and efficiency

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.