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Optimal error estimates for the $$hp$$-version interior penalty discontinuous Galerkin finite element method. (English) Zbl 1069.65118
Linear elliptic diffusion reaction problems in the plane with mixed Dirchlet and natural Neumann boundary conditions are considered. Discontinuous discretization with quadrilateral elements are investigated in detail. First, some recent results from approximation theory are recalled and applied to derive sharp estimates for bounds for norms of projection operators. These bounds finally are used to obtain optimal convergence estimates for the considered discontinuous $$hp$$-Galerkin method.

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