Automated adaptive error control in finite element methods using the error representation as error indicator
2014 (English)Report (Other academic)
In this paper we present a new adaptive finite element method directly using the a posteriori error representation as a local error indicator, and representing the primal and dual solutions in the same finite element space (here piecewise continuous linear functions on the same mesh). Since this approach gives a global a posteriori error estimate that is zero (due to Galerkin orthogonality), the error representation has traditionally been thought to contain no information about the error. However, we show the opposite, that locally, the orthogonal error representation behaves very similar to the non-orthogonal error representation using a higher order approximation of the dual, which is a standard approach to overcome the problem of a zero error estimate. We present evidence of this both in the form of an a priori estimate for the local error indicator for an elliptic model problem and a detailed computational investigation showing that the two methods exhibit very similar behavior and performance, and thus confirming the theoretical prediction. We also present computational results using a stabilized version of the method for non-elliptic partial differential equations where the error representation is no longer orthogonal, and where both the local error indicator and global error estimate behave similar to the error representation using a higher order approximation of the dual. The benefits of this adaptive method are generality and simplicity in formulation, sharpness, and efficiency since high order approximation of the dual and computation of additional constructs such as jump terms over interior facets or local problems are avoided.
Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2014. , 21 p.
CTL Technical Report
FEM adaptivity stabilized
IdentifiersURN: urn:nbn:se:kth:diva-146847OAI: oai:DiVA.org:kth-146847DiVA: diva2:725764
QC 201504172014-06-172014-06-172015-04-17Bibliographically approved