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An a posteriori error estimator for \(hp\)-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. (English) Zbl 1257.65062
The authors develop a residual-based a posteriori error estimator for the \(hp\)-symmetric interior penalty discontinuous Galerkin method in order to perform the discretization of the Laplace eigenvalue problem with homogeneous Dirichlet boundary conditions on bounded 2D and 3D domains. The reliability and the efficiency of the estimator are shown up to higher-order terms. Analogous error estimators can be obtained for more complicated elliptic eigenvalues problems. The numerical experiments which validate the theoretical results are on a square domain, an L-shaped domain, an H-shaped domain, a 3D cube and a circular domain. Under an \(hp\)-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenvalues.

65N15 Error bounds for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems 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
35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI
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