Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

Feistauer, Miloslav; Felcman, J.; Lukáčová-Medviďová, Mária; Warnecke, Gerald

doi:10.1137/s0036142997314695

Cited by 51 publications

(33 citation statements)

References 25 publications

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“…Some techniques such as using (asymptotic) Sobolev's inequalities and the bootstrap argument play a crucial role in the proof. A similar proof was presented in [17,21]. By (2.8), (2.9), and Lemma 3.4, we can derive the following H 1 and L 2 error estimates for the finite volume element scheme.…”

Section: (η T Isupporting

confidence: 57%

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Postprocessing of a finite volume element method for semilinear parabolic problems

Yang¹,

Bi²,

Liu³

2009

ESAIM: M2AN

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Abstract.In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L 2 and H 1 norms for the standard finite volume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency of the procedure.Mathematics Subject Classification. 65N30, 65N15.

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Section: (η T Isupporting

confidence: 57%

“…More importantly, the methods ensure local mass conservation, a highly desirable property in many applications. We refer to the monographs [16,22] for general presentations of these methods, and to the papers [2,4,5,11,17,27,32,33] (also the references therein) for more details.…”

Section: Introductionmentioning

confidence: 99%

Postprocessing of a finite volume element method for semilinear parabolic problems

Yang¹,

Bi²,

Liu³

2009

ESAIM: M2AN

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“…Discretizing this equation by the combined FE-FV scheme described above, with a rather general numerical flux adapted to the nonlinearity, and with a semi-implicit Euler method as time discretization, they derived L 2 (H 1 )-and L ∞ (L 2 )-error estimates. References [5,24,25,28] present results analogous to those in [2,18], but for a combined FE-FV method involving piecewise linear conforming finite elements and dual finite volumes (triangular finite volumes in the case of [5]). Similar L 2 (H 1 )-and L ∞ (L 2 )-error estimates as in [18] are shown in [27,50], but with respect to various discontinuous Galerkin schemes.…”

Section: Introductionmentioning

confidence: 85%

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

Deuring

Eymard

2017

ESAIM: M2AN

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Abstract.We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element -finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L 2 -stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.Mathematics Subject Classification. 65M12, 65M60.

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“…Fourthly, our results are valid for a larger family of conforming locally conservative discretizations, in the framework of the so-called combined finite volume-finite element method, cf. Feistauer et al [14] and the references therein. Consequently, the analysis includes such features as use of mass lumping for the time evolution or reaction terms.…”

Section: Introductionmentioning

confidence: 98%

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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Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

Cited by 51 publications

References 25 publications

Postprocessing of a finite volume element method for semilinear parabolic problems

Postprocessing of a finite volume element method for semilinear parabolic problems

L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

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

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Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

Cited by 51 publications

References 25 publications

Postprocessing of a finite volume element method for semilinear parabolic problems

Postprocessing of a finite volume element method for semilinear parabolic problems

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

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

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L²-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions