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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-022-01334-8