Strong Lp Convergence Associated with Rellich-type Discrete Compactness for Discontinuous Galerkin FEM

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 L2 convergence of some subfamily of finite element functions bounded in an H1-like mesh-dependent norm. In this note, we will show the strong Lp convergence of the above subfamily for 1 ≦ p < ∞. 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.