An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element

Abstract

We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates for the energy norm and for the L${}^2$ norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.

20 pages, 15 figures

References in corpus (4)

• Recovered Finite Element Methods
• A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem
• A finite element method by patch reconstruction for the Stokes problem using mixed formulations
• Solving Eigenvalue Problems in a Discontinuous Approximation Space by Patch Reconstruction

Cited by in corpus (6)

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• A Sequential Least Squares Method for Poisson Equation using A Patch Reconstructed Space
• Solving Eigenvalue Problems in a Discontinuous Approximation Space by Patch Reconstruction
• A Discontinuous Galerkin Method for the Stokes Equation by Divergence-free Patch Reconstruction
• A discontinuous Galerkin method by patch reconstruction for convection-diffusion-reaction problems over polytopic meshes