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Energy norm a posteriori error estimation of \(hp\)-adaptive discontinuous Galerkin methods for elliptic problems. (English) Zbl 1116.65115
The authors extend the technique proposed by P. Houston, D. Schötzau and T. Wihler [J. Sci. Comput. 22–23, 347–370 (2005; Zbl 1065.76139)] to the \(hp\)-version of the discontinuous Galerkin (DG) method and derive reliable upper bounds on the error measured in terms of a natural (mesh-dependent) energy norm for the DG approximation of the elliptic boundary-value problem. A posteriori error bounds are presented and discussed, both upper and lower energy norm bounds are derived. A series of numerical experiments to illustrate the performance of the proposed error estimators within an automatic \(hp\)-mesh refinement algorithm is presented.

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
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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