Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Pousin, Jérôme; Rappaz, Jacques

doi:10.1007/s002110050088

Cited by 88 publications

(80 citation statements)

References 10 publications

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“…This error estimator essentially consists of the elementwise error of the finite element functions with respect to the strong form of the differential equation and of jumps across inter-element boundaries of that boundary operator which naturally links the strong and weak forms of the differential equation. Some of the results of § §6-8 are completely new, others are generalizations of, and improvements upon, results previously obtained in [4,5,7,8,9,19,25,27,28]. We then trivially have a-l(\\F(u)\\Y-) < ||" -"oik < Q-x(\\F(u)\\Y.)…”

Section: Introductionsupporting

confidence: 62%

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A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Verfürth¹

1994

Math. Comp.

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Abstract. We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.

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Section: Introductionsupporting

confidence: 62%

“…(1) residual estimates: Estimate the error of the computed numerical solution by a suitable norm of its residual with respect to the strong form of the differential equation (cf., e.g., [4,5,9,19,21,25,27]). …”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

Verfürth¹

1994

Math. Comp.

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“…The extension of these results to the full nonlinear time-dependent Navier-Stokes equations has been considered and similar results can be proven, at least in the two-dimensional case, by using the abstract result in [13] (Th. 3), see also [15] (Prop.…”

Section: Introductionmentioning

confidence: 60%

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

Bernardi¹,

Verfürth²

2004

ESAIM: M2AN

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Abstract. The time-dependent Stokes equations in two-or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.Mathematics Subject Classification. 65N30, 65N15, 65J15.

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“…ii) Proof of the a-priori estimate (26). The L 2 estimate (26) is an immediate consequence of the H 1 estimate (25) and (73) in Lemma 6.…”

Section: Finally the Triangle Inequality Umentioning

confidence: 94%

“…Remark 2 Since the tensor a(x, s) depends on x, and also is not proportional in general to the identity I, the classical Kirchhoff transformation (see for instance [26]) cannot be used in our study.…”

Section: Remarkmentioning

confidence: 99%

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

Abdulle

Vilmart

2011

Numer. Math.

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To cite this version:Assyr Abdulle, Gilles Vilmart. A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems. Numerische Mathematik, Springer Verlag, 2012, 121, pp.397-431 Abstract The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.Keywords nonmonotone quasilinear elliptic problem · a priori error estimates · numerical quadrature · variational crime · finite elements.

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Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Cited by 88 publications

References 10 publications

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

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