Adaptive Finite Element Methods for Parabolic Problems IV: Nonlinear Problems

Eriksson, Kenneth; Johnson, Claes

doi:10.1137/0732078

Cited by 200 publications

(144 citation statements)

References 6 publications

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“…For elliptic and parabolic problems the theory of a posteriori error estimates of the form (1) is well developed [6], [7], [8], [9], [19]. But up to now there are no analogous results for initial value problems for nonlinear conservation laws of the form…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

1999

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Abstract. In this paper we shall derive a posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

1999

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“…The cG and dG time-stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3,5,10,11,16]. The application of cG and dG approaches to the time discretization of parabolic partial differential equations (PDEs) has been studied in [6][7][8][9]15,28] and the references therein.…”

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confidence: 99%

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

Schötzau

Wihler

2010

Numer. Math.

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The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.

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“…In this part we extend our studies to a posteriori estimates (see [1,2,[6][7][8][9]) dealing with the following basic problem: To construct an algorithm for the numerical solution of the coupling equations such that the error between the exact and approximate solutions, measured in some appropriate norms such as weighted L 2 (L 2 ) and L ∞ (H −1 ) norms, is guaranteed to be below a given tolerance and such that the computational cost is almost minimal. These two properties are referred to as the reliability and efficiency of the algorithm, respectively.…”

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confidence: 99%

A posteriori error estimates for the coupling equations of scalar conservation laws

Izadi

2009

Bit Numer Math

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In this paper we prove a posteriori L 2 (L 2 ) and L ∞ (H −1 ) residual based error estimates for a finite element method for the one-dimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ = 0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.

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Adaptive Finite Element Methods for Parabolic Problems IV: Nonlinear Problems

Cited by 200 publications

References 6 publications

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

A posteriori error estimates for the coupling equations of scalar conservation laws

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