Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

In this paper, we present a goal-oriented a posteriori error estimation technique for the pointwise error of finite element approximations using fundamental solutions. The approach is based on an integral representation of the pointwise quantity of interest using the corresponding Green's function, which is decomposed into an unknown regular part and a fundamental solution. Since only the regular part must be approximated with finite elements, very accurate results are obtained. The approach also allows the derivation of error bounds for the pointwise quantity, which are expressed in terms of the primal problem and the regular part problem. The presented technique is applied to linear elastic test problems in two-dimensions, but it can be applied to any linear problem for which fundamental solutions exist.

[1]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[2]  Friedel Hartmann,et al.  Introduction to Boundary Elements , 1989 .

[3]  Ekkehard Ramm,et al.  A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem , 1998 .

[4]  Harold A. Buetow,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[5]  Ian H. Sloan,et al.  Computable error bounds for pointwise derivatives of a Neumann problem , 1998 .

[6]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[7]  Anthony T. Patera,et al.  A hierarchical duality approach to bounds for the outputs of partial differential equations , 1998 .

[8]  Ivo Babuška,et al.  The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements , 1984 .

[9]  R. Rannacher,et al.  A feed-back approach to error control in finite element methods: application to linear elasticity , 1997 .

[10]  J. Oden,et al.  Goal-oriented error estimation and adaptivity for the finite element method , 2001 .

[11]  F. Hartmann,et al.  Duality and finite elements , 2004 .

[12]  F. Hartmann,et al.  Finite element recovery techniques for local quantities of linear problems using fundamental solutions , 2003 .

[13]  K. Bathe,et al.  Review: A posteriori error estimation techniques in practical finite element analysis , 2005 .

[14]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[15]  F. Hartmann.  The Mathematical Foundation of Structural Mechanics , 1985 .

[16]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[17]  Serge Prudhomme,et al.  On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , 1999 .

[18]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[19]  George F. Pinder,et al.  Analytical integration formulae for linear isoparametric finite elements , 1984 .

[20]  Ekkehard Ramm,et al.  A posteriori error estimation and adaptivity for elastoplasticity using the reciprocal theorem , 2000 .

[21]  D. Estep.  A posteriori error bounds and global error control for approximation of ordinary differential equations , 1995 .

[22]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[23]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .