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Computable finite element error bounds for Poisson's equation
Author(s) -
Barnhill R. R.,
Brown J. H.,
McQueen N.,
Mitchell A. R.
Publication year - 1977
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620110315
Subject(s) - mathematics , rectangle , finite element method , piecewise , poisson's equation , mathematical analysis , constant (computer programming) , convergence (economics) , galerkin method , geometry , physics , computer science , economics , thermodynamics , programming language , economic growth
Theoretical error bounds of the formare often available for finite element solutions U of elliptic problems. In this form the estimates give the order of convergence of the method but are of little practical value for estimating the size of the error because the magnitudes of the constant K and the theoretical solution u are unknown. An exception occurs in the case of the equation ∂ 2 u /∂ x 2 + ∂ 2 u /∂ y 2 + f = 0 in a rectangle where the Ritz–Galerkin finite element solution involves piecewise linears over a regular triangular grid. In this case where α = 1 andBarnhill and Gregory 1 have obtained the theoretical value 0·93√2 for K . In this note calculations are carried out for a variety of problems and the quantity K * = ∥ u – U ∥ E / h ∥ f ∥ L2 measured and compared with K . The values of K * obtained fit into a well defined pattern from which we conclude that the theoretical constant K is of the correct order of magnitude.