Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

Holst, Michael; Pollock, Sara

doi:10.1002/num.22002

Cited by 22 publications

(34 citation statements)

References 26 publications

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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%

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%

T-IFISS: a toolbox for adaptive FEM computation

Bespalov

Rocchi

Silvester

2021

Computers & Mathematics with Applications

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“…There exist several strategies of "combining" the two sets M u and M z into a single marking set that is used for refinement in the goal-oriented adaptive algorithm; see [MS09,BET11,HP16,FPZ16]. For goal-oriented adaptivity in the deterministic setting, [FPZ16] proves that the strategies of [MS09, BET11, FPZ16] lead to convergence with optimal algebraic rates, while the strategy from [HP16] might not. The marking strategy proposed in [FPZ16] is a modification of the strategy in [MS09].…”

Section: Marking Strategymentioning

confidence: 99%

Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs

Bespalov

Praetorius

Rocchi

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).Adaptive techniques based on rigorous a posteriori error analysis of computed solutions provide an effective mechanism for building approximation spaces and accelerating convergence of computed solutions. These techniques rely heavily on how the approximation error is estimated and controlled. One may choose to estimate the error in the global energy norm and use the associated error indicators to enhance the computed solution and drive the energy error estimate to zero. However, in practical applications, simulations often target a specific (e.g., localized) feature of the solution, called the quantity of interest and represented using a linear functional of the solution. In these cases, the energy norm may give very little useful information about the simulation error.Alternative error estimation techniques, such as goal-oriented error estimations, e.g., by the dual-weighted residual methods, allow to control the errors in the quantity of interest. While for deterministic PDEs, these error estimation techniques and the associated adaptive algorithms are very well studied (see, e.g., [EEHJ95, JS95, BR96, PO99, BR01, GS02, BR03] for the a posteriori error estimation and [MS09, BET11, HP16, FPZ16] for a rigorous convergence analysis of adaptive algorithms), relatively little work has been done for PDEs with parametric or uncertain inputs. For example, in the framework of (intrusive) stochastic Galerkin finite element methods (sGFEMs) (see, e.g., [GS91, LPS14]), a posteriori error estimation of linear functionals of solutions to PDEs with parametric uncertainty is addressed in [MLM07] and, for nonlinear problems, in [BDW11]. In particular, in [MLM07], a rigorous estimator for the error in the quantity of interest is derived and several adaptive refinement strategies are discussed. However, the authors comment that the proposed estimator lacks information about the structure of the estim...

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“…By the standard bound from (5) on the error in the goal-functional in terms of the primal and dual energy-norm errors [19,21,26]…”

Section: Online Basis Functionsmentioning

confidence: 99%

“…For example, in flow applications, one needs to obtain a good approximation of the pressure in locations where the wells are situated. Goal-oriented adaptivity [1,4,19,21,25,26,28,30,35] (and the references therein) can be used to more efficiently reduce the error in the quantity of interest without necessarily achieving the same rate of error reduction in a global sense. Goal-oriented adaptivity has been introduced within the setting of multiscale methodologies in for instance [2,3,29], where the authors review the framework of approximating a quantity of interest and investigate the use of this framework in a number of multiscale scientific applications (e.g.…”

Section: Introductionmentioning

confidence: 99%

Online basis construction for goal-oriented adaptivity in the generalized multiscale finite element method

Chung

Pollock

Pun

2019

Journal of Computational Physics

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In this research, we develop an online enrichment framework for goal-oriented adaptivity within the generalized multiscale finite element method for flow problems in heterogeneous media. The method for approximating the quantity of interest involves construction of residual-based primal and dual basis functions used to enrich the multiscale space at each stage of the adaptive algorithm. Three different online enrichment strategies based on the primal-dual online basis construction are proposed: standard, primal-dual combined and primal-dual product based. Numerical experiments are performed to illustrate the efficiency of the proposed methods for high-contrast heterogeneous problems.Keywords Goal-oriented adaptivity, multiscale finite element method, online basis construction, flow in heterogeneous media.

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Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

Cited by 22 publications

References 26 publications

T-IFISS: a toolbox for adaptive FEM computation

T-IFISS: a toolbox for adaptive FEM computation

Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs

Online basis construction for goal-oriented adaptivity in the generalized multiscale finite element method

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