Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

Angot, Philippe; Dolejší, Vít; Feistauer, Miloslav; Felcman, J.

doi:10.1023/a:1023217905340

Cited by 37 publications

(53 citation statements)

References 20 publications

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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: 87%

“…The one in [7] fits our situation best because in that latter reference, a Dirichlet boundary condition needs to hold only on part of the boundary of the domain in question. From ( [2], (3.30)) and (2.6), we obtain a formula for the…”

Section: Theorem 22 There Is a Constantmentioning

confidence: 99%

“…with the boundary conditions 2) and with the initial conditions w(x, 0) = w (0) (x) for x ∈ Ω, (1.3) constant of ζ. Moreover, there are some parameters entering linearly into this constant, namely the constants from certain Sobolev, interpolation and trace estimates on Ω.…”

Section: Introductionmentioning

confidence: 99%

“…[2,18], who considered a scalar time-dependent nonlinear conservation law with a diffusion term. 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.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 87%

Section: Theorem 22 There Is a Constantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…where ϕ Note that, for Crouzeix-Raviart elements, a combined finite volume/finite element method, similar to the technique employed here for the discretization of the momentum balance, has already been analysed for a transient non-linear convection-diffusion equation by Feistauer and co-workers [1,11,16].…”

Section: Space Discretization Of the Density Prediction And The Momenmentioning

confidence: 99%

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

Gallouët¹,

Gastaldo²,

Herbin³

et al. 2008

ESAIM: M2AN

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Abstract. We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2 -stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.Mathematics Subject Classification. 35Q30, 65N12, 65N30, 76M25.

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On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes

Feistauer

Slavk

Stupka

1999

Numer. Methods Partial Differential Eq.

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This article is a continuation of the work [M. Feistauer et al., Num Methods PDEs 13 (1997), 163-190] devoted to the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. In the previous article [1] the convergence of a semi-implicit scheme was established. Here we are concerned with the analysis of fully explicit schemes. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided that the mesh size tends to zero.

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Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

Cited by 37 publications

References 20 publications

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

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

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes

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Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

Cited by 37 publications

References 20 publications

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

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

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes

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

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