TY - JOUR
T1 - An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods
AU - Giani, Stefano
AU - Grubisic, Luka
AU - Hakula, Harri
AU - Ovall, Jeffrey S.
PY - 2018/2
Y1 - 2018/2
N2 - We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.
AB - We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.
KW - Eigenvalue problem
KW - Finite element method
KW - A posteriori error estimates
KW - Discontinuous Galerkin method
U2 - 10.1016/j.amc.2017.07.007
DO - 10.1016/j.amc.2017.07.007
M3 - Article
VL - 319
SP - 562
EP - 574
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
SN - 0096-3003
ER -