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Discontinuous Galerkin methods on \(hp\)-anisotropic meshes. II: A posteriori error analysis and adaptivity. (English) Zbl 1175.65127
Summary: We consider the a posteriori error analysis and \(hp\)-adaptation strategies for \(hp\)-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees.
In particular, we exploit duality based \(hp\)-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment.
The superiority of the proposed algorithm in comparison with standard \(hp\)-isotropic mesh refinement algorithms and an \(h\)-anisotropic/\(p\)-isotropic adaptive procedure is illustrated by a series of numerical experiments.
[For part I see Int. J. Comput. Sci. Math. 1, No. 2–3, 221–244 (2007)].

MSC:
65N15 Error bounds 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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