Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems

Alaoui, Linda El; Ern, Alexandre; Vohralı́k, Martin

doi:10.1016/j.cma.2010.03.024

Cited by 86 publications

(108 citation statements)

References 39 publications

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“…One approach consists in taking the dual norm of the residual, i.e., of the difference of the nonlinear operator applied to the exact and approximate solutions, cf. [20,37,32,43]. We also refer to [5,69] for the use of dual norms in singularly perturbed linear problems.…”

Section: Error Measurementioning

confidence: 99%

“…We also refer to [5,69] for the use of dual norms in singularly perturbed linear problems. The advantage is that such a measure is dictated by the problem at hand; it simplifies the analysis and leads to sharper (and possibly robust, as in [71,37,32,43]) estimates.…”

Section: Error Measurementioning

confidence: 99%

“…Following [49], [37], and [43], versions of (4.13a) and (4.13b) localized on each element of the mesh T n h should be used whenever one intends to refine adaptively the meshes T Moreover, it can be done completely in parallel. In practice, the estimators maybe only evaluated always after several iterations and various computational simplifications may be devised.…”

Section: Remark 43 (Local Stopping Criteria)mentioning

confidence: 99%

“…Inexact Newton methods have been studied in, e.g., [11,59,35,17,36,30,29]. Herein, we build upon the results of [49,42,37,47,43] which give guaranteed and robust a posteriori estimates with error components distinction.…”

Section: Introductionmentioning

confidence: 96%

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A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

Vohralı́k

Wheeler

2013

Comput Geosci

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International audienceThis paper develops a general abstract framework for a posteriori estimates for immiscible incompressible two-phase flows in porous media. We measure the error by the dual norm of the residual and, for mathematical correctness, employ the concept of global and complementary pressures in the analysis. Our estimators allow to estimate separately the different error components, namely the spatial discretization error, the temporal discretization error, the linearization error, the iterative coupling error, and the algebraic solver error. We propose an adaptive algorithm wherein the different iterative procedures (iterative linearization, iterative coupling, iterative solution of linear systems) are stopped when the corresponding errors do not affect significantly the overall error, and wherein the spatial and temporal errors are equilibrated. Consequently, important computational savings can be achieved while guaranteeing a user-given precision. The developed framework covers fully implicit, implicit pressure-explicit saturation, or iterative coupling formulations; conforming spatial discretization schemes such as the vertex-centered finite volume method or the finite element method, and nonconforming spatial discretization schemes such as the cell-centered finite volume method, the mixed finite element method, or the discontinuous Galerkin method; linearizations such as the Newton or the fixed-point one; and general linear solvers. Numerical experiments for a model problem are presented to illustrate the theoretical results. Only by stopping timely the linear and nonlinear solvers, speed-ups by a factor between 10 and 20 in terms of the number of total linear solver iterations are achieved

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Section: Error Measurementioning

confidence: 99%

Section: Error Measurementioning

confidence: 99%

Section: Remark 43 (Local Stopping Criteria)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

Vohralı́k

Wheeler

2013

Comput Geosci

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“…A posteriori error estimation for nonconforming finite element methods with application to second-order elliptic problems has recently seen significant progress and is still the subject of sustained research efforts, see e.g. [9,34,51,55,33,1,18,42,41,27,24] for dG methods and [16,35,2,17,37] for mixed finite element methods. We also refer the reader to [36,18,3,28,22] and references therein for the presentation of unifying frameworks on the topic.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Mozolevski

Prudhomme

2015

Computer Methods in Applied Mechanics and Engineering

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We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dualweighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed RaviartThomas method are used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

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A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

Dolejší

2013

Numerical Methods in Fluids

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SUMMARY We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.

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Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems

Cited by 86 publications

References 39 publications

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

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