A convergent FEM-DG method for the compressible Navier–Stokes equations

Karper, Trygve K.

doi:10.1007/s00211-013-0543-7

Cited by 86 publications

(73 citation statements)

References 25 publications

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“…The correct handling of coupling conditions in the network context, however, seems not straight forward; see [3] for analytical reasons. A globally convergent nonconforming finite element for the compressible Navier-Stokes equations has been proposed and analyzed recently [9,12,13]. Due to the somewhat unusual form of the coupling conditions, a direct generalization of this method to pipe networks again seems not feasible.…”

Section: Introductionmentioning

confidence: 99%

A Robust Conservative Mixed Finite Element Method for Isentropic Compressible Flow on Pipe Networks

Egger¹

2018

SIAM J. Sci. Comput.

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We consider the numerical approximation of compressible flow in a pipe network. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the conservation of mass and energy on the whole network. This variational principle, which is the basis of our further investigations, is amenable to a conforming Galerkin approximation by mixed finite elements. The resulting semi-discrete problems are well-posed and automatically inherit the global conservation laws for mass and energy from the continuous level. We also consider the subsequent discretization in time by a problem adapted implicit time stepping scheme which leads to conservation of mass and a slight dissipation of energy of the full discretization. The well-posedness of the fully discrete scheme is established and a fixed-point iteration is proposed for the solution of the nonlinear systems arising in every single time step. Some computational results are presented for illustration of our theoretical findings and for demonstration of the robustness and accuracy of the new method.

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Section: Introductionmentioning

confidence: 99%

A Robust Conservative Mixed Finite Element Method for Isentropic Compressible Flow on Pipe Networks

Egger¹

2018

SIAM J. Sci. Comput.

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“…Motivated by Karper [13], we use a mixed finite element finite volume method, where the convective terms are approximated by the standard upwind operator, while the diffusive term in the momentum equation is handled by means of the discontinuous Galerkin method based on the nonconformal finite elements of Crouzeix-Raviart-type. Accordingly, we consider an unfitted tetrahedral mesh generating a family of numerical domains {Ω h } h>0 such that (see section 2.2.1 for details)…”

Section: Numerical Analysismentioning

confidence: 99%

“…see Karper [13,Lemma 2.11]. Moreover, as a direct consequence of the shape regularity of the mesh, we record the error estimates…”

Section: Piecewise Linear Finite Elements We Start By Introducing Thmentioning

confidence: 99%

A Convergent Numerical Method for the Full Navier--Stokes--Fourier System in Smooth Physical Domains

Feireisl¹,

Hošek²,

Michálek³

2016

SIAM J. Numer. Anal.

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We propose a mixed finite volume-finite element numerical method for solving the full Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluid. The physical domain occupied by the fluid has a smooth boundary and it is approximated by a family of polyhedral numerical domains. Convergence and stability of the numerical scheme is studied. The numerical solutions are shown to converge, up to a subsequence, to a weak solution of the problem posed on the limit domain.

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“…Related results on the numerical approximation of compressible fluids employing the weak compactness tools developed by of Lions [23] in the discrete setting have been established by Karper et al [19,16,17,18] and Gallouët et al [13].…”

Section: 22mentioning

confidence: 99%

A convergent explicit finite difference scheme for a mechanical model for tumor growth

Trivisa

Weber

2016

ESAIM: M2AN

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Abstract. Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on imaging analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy's law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions [23].

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A convergent FEM-DG method for the compressible Navier–Stokes equations

Cited by 86 publications

References 25 publications

A Robust Conservative Mixed Finite Element Method for Isentropic Compressible Flow on Pipe Networks

A Robust Conservative Mixed Finite Element Method for Isentropic Compressible Flow on Pipe Networks

A Convergent Numerical Method for the Full Navier--Stokes--Fourier System in Smooth Physical Domains

A convergent explicit finite difference scheme for a mechanical model for tumor growth

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