Combined finite element -- finite volume method (convergence analysis)
Lukáčová-Medviďová, Mária
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997), p. 717-741 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We present an efficient numerical method for solving viscous compressible fluid flows. The basic idea is to combine finite volume and finite element methods in an appropriate way. Thus nonlinear convective terms are discretized by the finite volume method over a finite volume mesh dual to a triangular grid. Diffusion terms are discretized by the conforming piecewise linear finite element method. In the paper we study theoretical properties of this scheme for the scalar nonlinear convection-diffusion equation. We prove the convergence of the numerical solution to the exact solution.

Publié le : 1997-01-01
Classification:  35K60,  65M12,  65M60,  76M10,  76M25,  76N10 
@article{118968,
     author = {M\'aria Luk\'a\v cov\'a-Medvi\v dov\'a},
     title = {Combined finite element -- finite volume method (convergence analysis)},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {38},
     year = {1997},
     pages = {717-741},
     zbl = {0938.65111},
     mrnumber = {1603702},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118968}
}

Lukáčová-Medviďová, Mária. Combined finite element -- finite volume method (convergence analysis). Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 717-741. http://gdmltest.u-ga.fr/item/118968/

Adam D.; Felgenhauer A.; Roos H.-G.; Stynes M. A nonconforming finite element method for a singularly perturbed boundary value problem, Computing 54 1 (1995), 1-25. (1995) | MR 1314953 | Zbl 0813.65105

Bardos C.; Leroux A.Y.; Nedelec J.C. First order quasilinear equations with boundary conditions, Comm. in Part. Diff. Equa. 4 (9) (1979), 1017-1034. (1979) | MR 0542510

Berger H.; Feistauer M. Analysis of the finite element variational crimes in the numerical approximation of transonic flow, Math. Comput. 61 204 (1993), 493-521. (1993) | MR 1192967 | Zbl 0786.76051

Ciarlet P.G. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979. | MR 0520174 | Zbl 0547.65072

Feistauer M. Mathematical Methods in Fluid Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 67, Longman Scientific & Technical, Harlow, 1993. | MR 1266627 | Zbl 0819.76001

Feistauer M.; Felcman J.; Dolejší V. Adaptive finite volume method for the numerical solution of the compressible Euler equations, in: S. Wagner, E.H. Hirschel, J. Périaux and R. Piva, Eds., {Computational Fluid Dynamics 94}, Vol. 2, Proc. of the Second European CFD Conference (John Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapore, 1994), pp. 894-901.

Feistauer M.; Felcman J.; Lukáčová-Medviďová M. Combined finite element-finite volume solution of compressible flow, Journal of Comput. and Appl. Math. 63 (1995), 179-199. (1995) | MR 1365559

Feistauer M.; Felcman J.; Lukáčová-Medviďová M. On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems, Num. Methods for Part. Diff. Eqs. 13 (1997), 1-28. (1997) | MR 1436613

Feistauer M.; Knobloch P. Operator splitting method for compressible Euler and NavierStokes equations, Proc. of Internat. Workshop on Numerical Methods for the Navier-Stokes Equations, Heidelberg 1993, {Notes on Numerical Fluid Mechanics}, Vieweg, BraunschweigWiesbaden.

Felcman J. Finite volume solution of inviscid compressible fluid flow, ZAMM 71 (1991), 665-668. (1991)

Göhner U.; Warnecke G. A shock indicator for adaptive transonic flow computations, Num. Math. 66 (1994), 423-448. (1994) | MR 1254397

Göhner U.; Warnecke G. A second order finite difference error indicator for adaptive transonic flow computations, Num. Math. 70 (1995), 129-161. (1995) | MR 1324735

Ikeda T. Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Mathematics Studies 76, Lecture Notes in Numerical and Applied Analysis Vol. 4, North-Holland, Amsterdam-New York-Oxford, 1983. | MR 0683102 | Zbl 0508.65049

Lukáčová-Medviďová M. Numerical Solution of Compressible Flow, PhD Thesis, Fac. of Math. and Physics, Charles Univ., Prague, 1994.

Málek J.; Nečas J.; Rokyta M.; Růžička M. Weak and Measure Valued Solutions to Evolutionary Partial Differential Equations, Applied Mathematics and Mathematical Computation 13, London, Chapman & Hall, 1996.

Ohmori K.; Ushijima T. A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations, RAIRO Numer. Anal. 18 (1984), 309-322. (1984) | MR 0751761 | Zbl 0586.65080

Risch U. An upwind finite element method for singularly perturbed elliptic problems and local estimates in the $L^\infty$-norm, M$^{2}$AN 24 (1990), 235-264. (1990) | MR 1052149

Schieweck F.; Tobiska L. A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation, M$^{2}$AN 23 (1989), 627-647. (1989) | MR 1025076 | Zbl 0681.76032

Temam R. Navier-Stokes Equations, North-Holland, Amsterdam-New York-Oxford, 1977. | MR 0609732 | Zbl 1157.35333

Tobiska L. Full and weighted upwind finite element methods, in: Splines in Numerical Analysis (Mathematical Research Volume 52, J.W. Schmidt, H. Späth - eds.), Akademie-Verlag, Berlin, 1989. | MR 1004263 | Zbl 0685.65074

Vijayasundaram G. Transonic flow simulation using an upstream centered scheme of Godunov in finite elements, J. Comp. Phys. 63 (1986), 416-433. (1986) | MR 0835825