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Exponential convergence of the \(hp\)-DGFEM for diffusion problems. (English) Zbl 1059.65095
Two different formulations of the \(hp\)-discontinuous Galerkin finite element method (DGFEM) are considered for the two-dimensional stationary diffusion problem. As in the usual FEM, mesh refinement strategy is important when corner singularity caused by polygonal shape of domains exists. The authors prove exponential convergence of the \(hp\)-version of DGFEM on geometrically refined meshes in polygons. Several variants of interior penalization are covered. Numerical experiments indicate the sharpness of the theoretical results as well as the weak dependence of the DGFEM approximation on the particular choice of interior penalization and the penalty parameter. In certain cases, stabilization techniques are effective.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin 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
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