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

Abstract 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 ‖∇ u − ∇ w v h ‖ 0 is incorporated in the CDG formulation for second order elliptic equation, while for the case of fourth order equation, this term becomes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo stretchy="false">‖</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi mathvariant="double-struck">u</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mrow> <m:mi>w</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">v</m:mi> </m:mrow> <m:mrow> <m:mi>h</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mo stretchy="false">‖</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{v}}_{h}{{\Vert}}_{0}$ . These terms disappear if we directly analyze the nonstandard error formulations ‖∇ u − ∇ w u h ‖ 0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo stretchy="false">‖</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi mathvariant="double-struck">u</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mrow> <m:mi>w</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">u</m:mi> </m:mrow> <m:mrow> <m:mi>h</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mo stretchy="false">‖</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${\Vert}{\Delta}\mathbb{u}-{{\Delta}}_{w}{\mathbb{u}}_{h}{{\Vert}}_{0}$ . Extensive numerical results are also carried out to validate our theoretical findings.