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Convergence analysis of an adaptive interior penalty discontinuous Galerkin method. (English) Zbl 1189.65274
The authors discuss the convergence analysis of an adaptive interior penalty discontinuous Galerkin (IPDG) method for a second order 2D elliptic boundary value problem. Based on a residual-type a posteriori error estimate, they show that after each refinement step of the adaptive scheme, they achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the IPDG method. The convergence analysis does not require multiple interior nodes for refined elements of the triangulation. It is shown that the bisection of the elements is sufficient.
The main ingredient of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. Numerical results are provided to illustrate the performance of the adaptive method.

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
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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