# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
ARPACK
Full Text:
##### References:
 [1] DOI: 10.1016/S0045-7825(99)00242-X · Zbl 0956.65017 [2] Antonietti P., J. Comput. Appl. Math. 204 pp 317– [3] DOI: 10.1137/S0036142901384162 · Zbl 1008.65080 [4] DOI: 10.1090/S0025-5718-1989-0962210-8 [5] DOI: 10.1017/S0962492910000012 · Zbl 1242.65110 [6] DOI: 10.1137/080731918 · Zbl 1211.37094 [7] Descloux J., RAIRO Anal. Numér. 12 pp 97– · Zbl 0393.65024 [8] Descloux J., RAIRO Anal. Numér. 12 pp 113– [9] DOI: 10.1142/S0218202503002878 · Zbl 1072.65144 [10] DOI: 10.1142/S0218202509003590 · Zbl 1184.65100 [11] DOI: 10.1137/070697264 · Zbl 1191.65147 [12] DOI: 10.1016/j.apnum.2011.10.007 · Zbl 1236.65141 [13] Grisvard P., Singularities in Boundary Value Problems (1992) · Zbl 0766.35001 [14] DOI: 10.1090/S0025-5718-08-02181-9 · Zbl 1198.65221 [15] Hackbusch W., Elliptic Differential Equations (2003) [16] DOI: 10.1023/A:1014291224961 · Zbl 0995.65111 [17] DOI: 10.1142/S0218202507001826 · Zbl 1116.65115 [18] DOI: 10.1016/j.cma.2004.04.009 · Zbl 1074.65131 [19] DOI: 10.1093/imanum/drm009 · Zbl 1144.65070 [20] DOI: 10.1137/S0036142902405217 · Zbl 1058.65120 [21] DOI: 10.1137/1.9780898719628 · Zbl 0901.65021 [22] DOI: 10.1006/jcph.1998.6032 · Zbl 0926.65109 [23] DOI: 10.1023/A:1015118613130 · Zbl 1001.76060 [24] DOI: 10.1090/S0025-5718-02-01471-0 · Zbl 1084.78007 [25] Schwab C., p- and hp-Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics (1999) [26] Solin P., Higher Order Finite Element Methods (2003) [27] Verfürth R., A Review of Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (1996) · Zbl 0853.65108 [28] DOI: 10.1016/j.cma.2006.10.036 · Zbl 1173.74446 [29] DOI: 10.1142/S0218202511005052 · Zbl 1220.65156 [30] DOI: 10.1093/imanum/drp038 · Zbl 1225.65104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.