An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element
This work addresses the high computational cost of standard discontinuous Galerkin methods by drastically reducing degrees of freedom, benefiting large-scale simulations on polygonal meshes.
The paper proposes a discontinuous Galerkin method for second-order elliptic problems on polygonal meshes that uses only one degree of freedom per element, achieving optimal error estimates. Numerical tests up to order six demonstrate accuracy and efficiency.
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.