An Abstract Analysis of Optimal Goal-Oriented Adaptivity

Feischl, Michael; Praetorius, Dirk; Zee, Kristoffer G. van der

doi:10.1137/15m1021982

Cited by 44 publications

(71 citation statements)

References 44 publications

(220 reference statements)

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“…and µ E (resp., ζ E ) is the local contribution to µ (resp., ζ) associated with the edge E; (GO-MARK4) this marking strategy is a modification of (GO-MARK2); following [27], we compare the cardinality of M u and that of M z to define Comparing these four strategies, it is proved in [27,Theorem 13] that the GOAFEM algorithm employing marking strategies (GO-MARK2)-(GO-MARK4) generates approximations that converge with optimal algebraic rates, whereas only suboptimal convergence rates have been proved for marking strategy (GO-MARK1); cf. [27,Remark 4] and [34,Section 4]. The numerical results in [27] suggest that (GO-MARK4) is more effective than the original strategy (GO-MARK2) in terms of the overall computational cost.…”

Section: Adaptive Finite Element Methods (Fem)mentioning

confidence: 99%

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T-IFISS: a toolbox for adaptive FEM computation

Bespalov

Rocchi

Silvester

2021

Computers & Mathematics with Applications

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T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration, enabling users to quickly develop new discretizations and test alternative algorithms. The package is also valuable as a teaching tool for students who want to learn about state-of-the-art finite element methodology. and is compatible with Windows, Linux and MacOS computers.3 We would almost certainly use an iterative solver preconditioned with an algebraic multigrid V-cycle if we were trying to solve the same PDE problem in three spatial dimensions. 4 The default uniform refinement in T-IFISS is by three bisections (see Figure 1(d)). However, there is an option to switch to the so-called red uniform refinement (i.e., the one obtained by connecting the edge midpoints of each triangle); this can be done by setting subdivPar = 1 within the function adiff adaptive main.m.

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Section: Adaptive Finite Element Methods (Fem)mentioning

confidence: 99%

“…[27,Remark 4] and [34,Section 4]. The numerical results in [27] suggest that (GO-MARK4) is more effective than the original strategy (GO-MARK2) in terms of the overall computational cost. Our own experience is that (GO-MARK4) is a competitive strategy in every example that has been tested.…”

Section: Adaptive Finite Element Methods (Fem)mentioning

confidence: 99%

T-IFISS: a toolbox for adaptive FEM computation

Bespalov

Rocchi

Silvester

2021

Computers & Mathematics with Applications

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“…, 9 as representative parameters. , [FPZ16], and [BET11] (left to right) with polynomial degree p = 1 (top) and p = 2 (bottom), and marking parameter ϑ = 0.5. The colors show the value of log 2 (1/|T |) for every element T , which corresponds to the element's level.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…see, e.g., [MS09,FPZ16]. Therefore, GOAFEM aims to control and steer the product of the total errors…”

Section: Introductionmentioning

confidence: 99%

Instance-Optimal Goal-Oriented Adaptivity

Innerberger

Praetorius

2020

Computational Methods in Applied Mathematics

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We consider an adaptive finite element method with arbitrary but fixed polynomial degree p ≥ 1, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [Diening et al, Found. Comput. Math. 16, 2016], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. Numerical experiments underline our theoretical findings.Date: July 31, 2019. 2010 Mathematics Subject Classification. 65N30, 41A25, 65N12, 65N50. Key words and phrases. Adaptive finite element method, goal-oriented algorithm, quantity of interest, maximum marking strategy, convergence, instance optimality.Acknowledgement: The authors acknowledge support through the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), the special research program Taming complexity in PDE systems (grant SFB F65), and the stand-alone project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).

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“…There exist numerous engineering applications that motivate the use of GOA, including electromagnetics, 4-8 structural problems and visco-elasticity, [9][10][11][12][13] fluid-structure interactions, [14][15][16] and control theory. [17][18][19] Apart from these applications, convergence properties of GOA have also been recently studied in Pollock, Holst et al, Holst and Pollock, Mommer and Stevenson, and Feischl et al [20][21][22][23][24] The origin of the GOA is in the works of Rannacher et al [25][26][27] followed by the works of Peraire; Patera et al [28][29][30][31][32][33] on a posteriori error estimates of the error in the QoI. The works of Prudhomme and Oden [34][35][36][37] formulate the goal-oriented error estimation procedure based on representing the error in the QoI in terms of global functions defined over the entire computational domain.…”

Section: Introductionmentioning

confidence: 99%

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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SummaryIn goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity.While the method can be applied to a variety of problems, we focus here on two-and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

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An Abstract Analysis of Optimal Goal-Oriented Adaptivity

Cited by 44 publications

References 44 publications

T-IFISS: a toolbox for adaptive FEM computation

T-IFISS: a toolbox for adaptive FEM computation

Instance-Optimal Goal-Oriented Adaptivity

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

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