Strong $L^p$ convergence associated with Rellich-type discrete compactness for discontinuous Galerkin FEM

Abstract

In a preceding paper, we proved the discrete compactness properties of Rellich type for some 2D discontinuous Galerkin finite element methods (DGFEM), that is, the strong $L^2$ convergence of some subfamily of finite element functions bounded in an $H^1$-like mesh-dependent norm. In this note, we will show the strong $L^p$ convergence of the above subfamily for $1 \le p < \infty$. To this end, we will utilize the duality mappings and special auxiliary problems. The results are applicable to numerical analysis of various semi-linear problems.