The Quadratic Discontinuous Finite Volume Element Schemes for Elliptic Problems

Abstract

The objective of the paper is to develop the quadratic discontinuous finite volume methods (DFVM) for solving elliptic equations with variable coefficients. The proposed numerical schemes are composed of the discontinuous Galerkin (DG) method with different interior penalty formulations (IIPG, NIPG, SIPG) and the finite volume element method which the trial function is discontinuous quadratic element function. Subsequently, with the specialized projection techniques, we built up a bridge between the bilinear form of DFVM and that of DG method, which simplifies the analysis and proves the optimal error estimate of the schemes in the broken $H^1$ norm. Finally, we provide numerical simulations to validate the theoretical findings.