Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver l...

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Vydáno v:	Numerische Mathematik Ročník 153; číslo 1; s. 111 - 140
Hlavní autoři:	Becker, Roland, Gantner, Gregor, Innerberger, Michael, Praetorius, Dirk
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2023 Springer Nature B.V Springer Verlag
Témata:	Adaptive algorithms Complexity Computing costs Conjugate gradient method Convergence Finite element method Iterative methods Linear systems Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Optimization Simulation Theoretical 65N50 65Y20 41A25 65N22 65N30
ISSN:	0029-599X, 0945-3245
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Abstract	We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.
AbstractList	We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost. We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.
Author	Praetorius, Dirk Innerberger, Michael Becker, Roland Gantner, Gregor
Author_xml	– sequence: 1 givenname: Roland surname: Becker fullname: Becker, Roland organization: IPRA-LMAP, Université de Pau et des Pays de l’Adour – sequence: 2 givenname: Gregor surname: Gantner fullname: Gantner, Gregor organization: Korteweg-de Vries (KdV) Institute for Mathematics, University of Amsterdam – sequence: 3 givenname: Michael surname: Innerberger fullname: Innerberger, Michael email: michael.innerberger@asc.tuwien.ac.at organization: TU Wien, Institute of Analysis and Scientific Computing – sequence: 4 givenname: Dirk surname: Praetorius fullname: Praetorius, Dirk organization: TU Wien, Institute of Analysis and Scientific Computing
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Cites_doi	10.1016/j.apnum.2016.11.005 10.1137/07069047X 10.1090/S0025-5718-07-01959-X 10.1007/s00211-011-0401-4 10.1137/15M1021982 10.1007/s10208-005-0183-0 10.1137/120897225 10.1007/s00365-013-9192-4 10.1093/imanum/drr014 10.1137/0733054 10.1137/060675666 10.1016/j.camwa.2017.12.035 10.1137/100794298 10.1007/s00211-015-0727-4 10.1090/mcom/3553 10.1017/S0962492901000010 10.1016/j.camwa.2013.12.003 10.1093/imanum/drx050 10.1090/mcom/3654 10.1017/S0962492900002531 10.1137/S0036142999360044 10.1007/s00211-003-0492-7 10.1007/978-3-0348-7605-6 10.1017/S096249290200003X
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Snippet	We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which...
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SubjectTerms	Adaptive algorithms Complexity Computing costs Conjugate gradient method Convergence Finite element method Iterative methods Linear systems Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Optimization Simulation Theoretical
Title	Goal-oriented adaptive finite element methods with optimal computational complexity
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