Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

Dolejší, Vít; Feistauer, Miloslav; Felcman, J.; Kliková, A.

doi:10.1023/a:1021701705932

Cited by 18 publications

(28 citation statements)

References 35 publications

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“…combining a Crouzeix-Raviart finite element discretization for the diffusion and a finite volume discretization for the convection, are available in some cases: incompressible stationary flows [10,11], convectiondiffusion equations [7][8][9]. However, the only analysis for Navier-Stokes equations [10,11] is performed in the incompressible framework and for an upwind approximation for the convection, the resulting scheme being only first order in space.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Another approach, which consists of discretizing the convection term by a finite volume technique based on a dual mesh, has been proposed in the literature. It has been first applied for the Crouzeix-Raviart element to a scalar linear [6] and 557 nonlinear [7][8][9] convection diffusion equation, with the goal to obtain a monotone approximation of the convection operator and a general discretization of the diffusion (in particular, suitable for anisotropic diffusion). Then this technique has been extended to incompressible stationary flows, for the Crouzeix-Raviart element [10,11], and further for the Rannacher-Turek element [12, section 3.1.4, pp.…”

Section: Introductionmentioning

confidence: 99%

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An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

Ansanay-Alex¹,

Babik²,

Latché³

et al. 2011

Numerical Methods in Fluids

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SUMMARYWe develop in this paper a discretization for the convection term in variable density unstationary Navier-Stokes equations, which applies to low-order non-conforming finite element approximations (the so-called Crouzeix-Raviart or Rannacher-Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L 2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L 2 norm, second-order space convergence for the velocity and first-order space convergence for the pressure are observed.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

Ansanay-Alex¹,

Babik²,

Latché³

et al. 2011

Numerical Methods in Fluids

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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: 89%

“…[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%

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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“…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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Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

Cited by 18 publications

References 35 publications

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

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

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Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

Cited by 18 publications

References 35 publications

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

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

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