A medius error analysis for the conforming discontinuous Galerkin finite element methods

In this paper, we derive an improved error estimate of a conforming discontinuous Galerkin (CDG) method for both second and fourth order elliptic problems, assuming only minimal regularity on the exact solutions. The result we established is called a medius error estimate since it relies on both a priori and a posteriori analysis. Compared with the standard interior penalty discontinuous Galerkin (IPDG) method, when choosing the standard DG norm, an additional term & Vert;del u - del w v h & Vert;0 is incorporated in the CDG formulation for second order elliptic equation, while for the case of fourth order equation, this term becomes & Vert; Delta u - Delta w v h & Vert; 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{v}}_{h}{{\Vert}}_{0}$ . These terms disappear if we directly analyze the nonstandard error formulations & Vert;del u - del w u h & Vert;0 and & Vert; Delta u - Delta w u h & Vert; 0 ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{u}}_{h}{{\Vert}}_{0}$ . Extensive numerical results are also carried out to validate our theoretical findings.