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

TL;DR

This paper introduces a novel high-order discontinuous Galerkin method for second-order elliptic problems on polygonal meshes, using only one unknown per element, and demonstrates its optimal error estimates and efficiency.

Contribution

It presents a new arbitrary-order DG method with a single unknown per element, utilizing local least-squares solutions on patches, with proven optimal error estimates.

Findings

Achieves optimal error estimates for energy and L2 norms.

Demonstrates high accuracy and efficiency up to order six.

Validates method performance on various polygonal meshes.

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.