We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space-time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space-time finite element method).
%0 Journal Article
%1 kreuzer2012design
%A Kreuzer, Christian
%A Möller, Christian A.
%A Schmidt, Alfred
%A Siebert, Kunibert G.
%D 2012
%I Oxford Univ. Press
%J IMA journal of numerical analysis
%K liste ubs_10008 ubs_20013 ubs_30123 unibibliografie
%N 4
%P 1375-1403
%R 10.1093/imanum/drr026
%T Design and Convergence Analysis for an Adaptive Discretization of the Heat Equation
%V 32
%X We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space-time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space-time finite element method).
@article{kreuzer2012design,
abstract = {We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space-time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space-time finite element method).},
added-at = {2020-03-27T18:54:40.000+0100},
author = {Kreuzer, Christian and Möller, Christian A. and Schmidt, Alfred and Siebert, Kunibert G.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/292bb967f604ba8afc9859daa25fce87d/unibiblio},
doi = {10.1093/imanum/drr026},
interhash = {8cb8952ea05b1a7da9074f13decd9459},
intrahash = {92bb967f604ba8afc9859daa25fce87d},
issn = {{0272-4979} and {1464-3642}},
journal = {IMA journal of numerical analysis},
keywords = {liste ubs_10008 ubs_20013 ubs_30123 unibibliografie},
language = {eng},
number = 4,
pages = {1375-1403},
publisher = {Oxford Univ. Press},
timestamp = {2021-06-18T14:51:00.000+0200},
title = {Design and Convergence Analysis for an Adaptive Discretization of the Heat Equation},
volume = 32,
year = 2012
}