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$$hp$$-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes. (English) Zbl 1102.65112
The $$hp$$-version of the interior penalty discontinuous Galerkin finite element method ($$hp$$-DGFEM) for linear second-order elliptic reaction-diffusion-advection equations is considered. The error analysis of the $$hp$$-DGFEM is extended to the case when anisotropic (shape-irregular) elements and anisotropic polynomial degrees are used.
For this purpose, results of P. Houston, C. Schwab and E. Süli [SIAM J. Numer. Anal. 39, 2133–2163 (2002; Zbl 1015.65067)] are extended to the anisotropic case. The choice for a user-defined discontinuity penalization parameter is not so easy on grids with anisotropic elements.

MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations