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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Published in:	Numerische Mathematik Vol. 153; no. 1; pp. 111 - 140
Main Authors:	Becker, Roland, Gantner, Gregor, Innerberger, Michael, Praetorius, Dirk
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2023 Springer Nature B.V Springer Verlag
Subjects:	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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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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