Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

Mallik, Gouranga; Vohralı́k, Martin; Yousef, Soleiman

doi:10.1016/j.cam.2019.112367

Cited by 27 publications

(20 citation statements)

References 46 publications

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“…Finally, Appendix C presents the application of our mass-conservative total flux reconstruction in H(div, Ω) to a challenging two-phase porous media flow problem with a finite volume fully implict/iterative coupling discretization. Applications to other problems, namely when deriving guaranteed upper bounds on the total error in presence of inexact solvers, have already been considered in [34,35] to steady and unsteady variational inequalities, in [26] to eigenvalue problems, in [60] to goal-oriented error estimates, and in [87,2] to degenerate multiphase (multicompositional) flows.…”

Section: Introductionmentioning

confidence: 99%

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

Papež

Rüde

Vohralı́k

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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We present novel H(div) and H 1 liftings of given piecewise polynomials over a hierarchy of simplicial meshes, based on a global solve on the coarsest mesh and on local solves on patches of mesh elements around vertices on subsequent mesh levels. This in particular allows to lift a given algebraic residual. In connection with approaches lifting the total residual, we show how to obtain guaranteed, fully computable, and constant-free upper and lower a posteriori bounds on the algebraic, total, and discretization errors; here we consider the model Poisson equation discretized by the conforming finite element method of arbitrary order and including an arbitrary iterative solver. We next formulate safe stopping criteria ensuring that the algebraic error does not dominate the total error. We also prove efficiency, i.e., equivalence of our upper total and algebraic estimates with the total and algebraic errors, respectively, up to a generic constant; this constant is polynomial-degree-independent for the total error. Numerical experiments illustrate sharp control of all error components and accurate prediction of their spatial distribution in several test problems, including cases where some classical estimators fail. The H(div)-liftings at the same time allow to recover mass balance for any problem, any numerical discretization, and any situation such as inexact solution of (nonlinear) algebraic systems or algorithm failure, which we believe is of independent interest. We demonstrate this mass balance recovery in a simulation of immiscible incompressible two-phase flow in porous media.

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Section: Introductionmentioning

confidence: 99%

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

Papež

Rüde

Vohralı́k

et al. 2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…In fact, it was observed in Reference 38 that g m is in most cases a better approximation of the true value g ex . A bounding of g ex is, therefore:

g^{-} \leq g_{e x} \leq g^{+},

where

g^{-} : = g^{m} - \frac{1}{2} e_{C R_{Ω}} ({\underline{u}}_{H}, {\hat{\underline{\underline{σ}}}}_{H}) e_{C R_{Ω}} ({\underline{ũ}}_{\tilde{H}}, {\hat{\underline{\underline{\tilde{σ}}}}}_{\tilde{H}}),

and

g^{+} : = g^{m} + \frac{1}{2} e_{C R_{Ω}} ({\underline{u}}_{H}, {\hat{\underline{\underline{σ}}}}_{H}) e_{C R_{Ω}} ({\underline{ũ}}_{\tilde{H}}, {\hat{\underline{\underline{\tilde{σ}}}}}_{\tilde{H}}) .

…”

Section: Settingsmentioning

confidence: 96%

Multifidelity adaptive kriging metamodel based on discretization error bounds

Mell

Rey

Schoefs

2020

Numerical Meth Engineering

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Summary This article presents an approach to build a multifidelity kriging metamodel from finite element computations on different meshes for stuctural reliability assessment. The proposed method takes advantage of the computation of bounds on the discretization error, which enables to guarantee the state (safe or failure) of each computation of the performance function. An algorithm to build the metamodel from the different levels of fidelity and estimate the failure probability is provided. Illustrations are presented on a two dimensional mechanical crack opening problem. Bounds on the failure probability are also post‐processed.

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“…Recent developments dedicated a great deal of effort to account for inexactness of the algebraic approximations and introduce stopping criteria based on the interplay between discretization and algebraic computation in adaptive FEM. Among others, we mention the seminal contributions [10,26,4,5,34,30,33,32,21,31,29,20].…”

Section: Prolongate Smooth Solvementioning

confidence: 99%

Quasi-Optimal Mesh Sequence Construction through Smoothed Adaptive Finite Element Methods

Mulita¹,

Giani²,

Heltai³

2021

SIAM J. Sci. Comput.

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We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost.We quantify rigorously how the error propagates throughout the algorithm, and we provide a connection with classical a posteriori error analysis. A series of numerical experiments highlights the efficiency and the computational speedup of S-AFEM.

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Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

Cited by 27 publications

References 46 publications

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

Multifidelity adaptive kriging metamodel based on discretization error bounds

Quasi-Optimal Mesh Sequence Construction through Smoothed Adaptive Finite Element Methods

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