An Adaptive Algorithm for Ordinary, Stochastic and Partial Differential Equations
2005 (English)In: Recent Advances in Adaptive Computation, Providence: American Mathematical Society , 2005, 325-343 p.Chapter in book (Other academic)
The theory of a posteriori error estimates suitable for adaptive refinement is well established. This work focuses on the fundamental, but less studied, issue of convergence rates of adaptive algorithms. In particular, this work describes a simple and general adaptive algorithm applied to ordinary, stochastic and partial differential equations with proven convergence rates. The presentation has three parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. It is explained how to measure optimal convergence rates for approximation of functionals of the solution, and why convergence of the error density is always useful and subtle in the case of stochastic and partial differential equations. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor.
Place, publisher, year, edition, pages
Providence: American Mathematical Society , 2005. 325-343 p.
, Contemporary mathematics, ISSN 0271-4132 ; 383
FINITE-ELEMENT METHODS; CONVERGENCE-RATES; ERROR; APPROXIMATION; ESTIMATORS
IdentifiersURN: urn:nbn:se:kth:diva-7420ISBN: 0-8218-3662-5OAI: oai:DiVA.org:kth-7420DiVA: diva2:12443
QC 201008242007-08-312007-08-312010-08-24Bibliographically approved