<b><i>A posteriori</i></b>error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Ohlberger, Mario

doi:10.1051/m2an:2001119

Cited by 83 publications

(81 citation statements)

References 38 publications

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“…For well-posedness of degenerate parabolic equations with Dirichlet boundary conditions we refer to [6], in the case of mixed Dirichlet-Neumann conditions to [19]. Convergence results and error estimates were derived in [11,27].…”

Section: Finite Volume Approximationmentioning

confidence: 99%

Reduced basis method for finite volume approximations of parametrized linear evolution equations

Haasdonk¹,

Ohlberger²

2008

ESAIM: M2AN

Self Cite

359

457

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Abstract.The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

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Section: Finite Volume Approximationmentioning

confidence: 99%

Reduced basis method for finite volume approximations of parametrized linear evolution equations

Haasdonk¹,

Ohlberger²

2008

ESAIM: M2AN

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359

457

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“…Discretization of the aforementioned hyperbolic, porous medium, convection-diffusion, and ellipticparabolic equations by finite volume methods is quite standard by now and often used in engeneering practice. We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8.…”

Section: Introductionmentioning

confidence: 99%

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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“…Let the meshes T n h be given first and let the meshes D n h be constructed using the face, edge, and element barycentres. Then the combined finite volume-finite element scheme (3.1) is equivalent to the classical vertex-centered finite volume method (cf., e.g., [19] and the references therein), where mass lumping has been used in the time evolution and reaction terms. This follows easily using [3,Lemma 3] for the diffusion term.…”

Section: Remark 33 (Arithmetic Versus Harmonic Averaging)mentioning

confidence: 99%

“…Fewer results are known in the unsteady case. L 1 -norm estimates for nonlinear problems are derived by Ohlberger [19], whereas the energy norm setting has been pursued in, e.g., Felcman and Kubera [15] or Amaziane et al [1]. Typically, the estimate only gives the error upper bound up to an undetermined constant, so that the actual error control is not possible.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Hilhorst¹,

Vohralı́k²

2011

Computer Methods in Applied Mechanics and Engineering

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We derive fully computable a posteriori error estimates for vertex-centered finite volume-type discretizations of transient convection-diffusion-reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. Moreover, this algorithm leads to optimal efficiency as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume-finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. Numerical experiments illustrate the theoretical developments.

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A posteriorierror estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Cited by 83 publications

References 38 publications

Reduced basis method for finite volume approximations of parametrized linear evolution equations

Reduced basis method for finite volume approximations of parametrized linear evolution equations

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

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