Guaranteed error bounds and local indicators for adaptive solvers using stabilised space–time IgA approximations to parabolic problems

Langer, Ulrich; Matculevich, Svetlana; Repin, Sergey

doi:10.1016/j.camwa.2019.04.009

Cited by 16 publications

(10 citation statements)

References 65 publications

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“…Higher-order B-splines of highest smoothness even on coaerser meshes can be used to construct enriched spaces U l,(2) h and V l,(2) h that lead to cheap problems on the enriched spaces, see [33,34,35] for the successful use of this technique in functional-type a posteriori error estimates.…”

Section: Spatial Discretizationmentioning

confidence: 99%

Multigoal-oriented error estimates for non-linear problems

Endtmayer

Langer

Wick

2019

Journal of Numerical Mathematics

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Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

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Section: Spatial Discretizationmentioning

confidence: 99%

Multigoal-oriented error estimates for non-linear problems

Endtmayer

Langer

Wick

2019

Journal of Numerical Mathematics

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“…In these studies, adaptivity is based on a posteriori error estimation, which is a widely used and well developed tool in finite element (FEM) computations as, for example, presented in [3,7,8,12,31,37,50,62,68,70], and in other discretization techniques too; see, e.g., [6,45,47,54,65,69].…”

Section: Introductionmentioning

confidence: 99%

Two-Side a Posteriori Error Estimates for the Dual-Weighted Residual Method

Endtmayer¹,

Langer²,

Wick³

2020

SIAM J. Sci. Comput.

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In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests.

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“…The universality of the functional error estimates allows to easily extend them to parabolic problems. The corresponding results on the application of these error bounds to space-time IgA techniques in a context of evolutionary problems can be found in [34].…”

Section: Resultsmentioning

confidence: 99%

Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

Matculevich

2018

Journal of Computational and Applied Mathematics

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This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundary-value problems. Along with the detailed discussion of the most crucial properties of such estimates, we present the algorithm of a reliable solution approximation together with the scheme of an efficient a posteriori error bound generation. In this approach, we take advantage of B-(THB-) spline's high smoothness for the auxiliary vector function reconstruction, which, at the same time, allows to use much coarser meshes and decrease the number of unknowns substantially. The most representative numerical results, obtained during a systematic testing of error estimates, are presented in the second part of the paper. The efficiency of the obtained error bounds is analysed from both the error estimation (indication) and the computational expenses points of view. Several examples illustrate that functional error estimates (alternatively referred to as the majorants and minorants of deviation from an exact solution) perform a much sharper error control than, for instance, residual-based error estimates. Simultaneously, assembling and solving routines for an auxiliary variables reconstruction, which generate the majorant (or minorant) of an error, can be executed several times faster than the routines for a primal unknown.

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Guaranteed error bounds and local indicators for adaptive solvers using stabilised space–time IgA approximations to parabolic problems

Cited by 16 publications

References 65 publications

Multigoal-oriented error estimates for non-linear problems

Multigoal-oriented error estimates for non-linear problems

Two-Side a Posteriori Error Estimates for the Dual-Weighted Residual Method

Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

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