Premium
Fully reliable error control in the h‐p‐version of FEM
Author(s) -
Carstensen C.,
Funken S. A.
Publication year - 1999
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19990791336
Subject(s) - a priori and a posteriori , finite element method , residual , eigenvalues and eigenvectors , mathematics , polygon mesh , partition (number theory) , constant (computer programming) , upper and lower bounds , mathematical optimization , algorithm , mathematical analysis , computer science , geometry , combinatorics , philosophy , physics , epistemology , quantum mechanics , thermodynamics , programming language
Some approaches in the a posteriori error analysis of finite element methods (FEM) are based on the regularity of the exact solution or on a saturation property of the numerical scheme. For coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds. Here, we will provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well. A localisation via a partition of unity yields problems on small domains. Two fully reliable estimates are established, the sharper one solves an analytical interface problem with residuals following Babugka and Rheinboldt. The second estimate yields a modification of the standard residual‐based a posteriori estimate with explicit constants computed from local analytical eigenvalue problems. Emphasis is on the efficiency of the computed error bound, which can be monitored. For some class of triangulations and the h‐version we show that the efficiency constant is smaller than 2.5 and grows only weakly for the h‐p‐version. Numerical experiments support and illustrate the theoretical results.