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Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems. (English) Zbl 0942.76035

The paper presents numerical analysis of a nonlinear convection-diffusion problem in a two-dimensional polygonal domain. A combined barycentric finite volume and nonconforming triangular piecewise-linear finite element method is used for the discretization of nonlinear convective terms and linear diffusion terms, respectively, while the time discretization is implicit. Under some standard assumptions, the authors prove the discrete maximum principle, and establish stability and consistency of the discretization. Some a priori error estimates allow to prove the convergence theorem. In the introduction, the authors mention selected applications of the approach to problems of fluid dynamics.
Reviewer: K.Segeth (Praha)

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

[1] D. Adam, A. Felgenhauer, H.-G. Roos and M. Stynes: A nonconforming finite element method for a singularly perturbed boundary value problem. Computing 54 (1995), 1-26. · Zbl 0813.65105
[2] D. Adam and H.-G. Roos: A nonconforming exponentially field fitted finite element method I: The interpolation error. Preprint MATH-NM-06-1993, Technische Universität Dresden, 1993.
[3] P. Arminjon, A. Dervieux, L. Fezoui, H. Steve and B. Stoufflet: Non-oscillatory schemes for multidimensional Euler calculations with unstructured grids. In Notes on Numerical Fluid Mechanics Volume 24 (Nonlinear Hyperbolic Equations-Theory, Computation Methods and Applications), Ballman J. and Jeltsch R. (eds.), Technical report, Vieweg, Braunschweig-Wiesbaden, 1989 1989, pp. 1-10. · Zbl 0661.65091
[4] P. G. Ciarlet: The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam, 1979.
[5] M. Crouzeix and P.-A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO, Anal. Numér. 7 (1973), 733-76. · Zbl 0302.65087
[6] V. Dolejší and P. Angot: Finite Volume Methods on Unstructured Meshes for Compressible Flows. In Finite Volumes for Complex Applications (Problems and Perspectives), F. Benkhaldoun and R. Vilsmeier (eds.), ISBN 2-86601-556-8, Hermes, Rouen, 1996, pp. 667-674.
[7] V. Dolejší, M. Feistauer and J. Felcman: On the discrete Friedrichs inequality for nonconforming finite elements. Preprint, Faculty of Mathematics and Physics, Charles University, Prague 1998. · Zbl 0938.65130
[8] M. Feistauer: Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Monographs and Surveys in Pure and Applied Mathematics 67, Harlow, 1993. · Zbl 0819.76001
[9] M. Feistauer and J. Felcman: Convection-diffusion problems and compressible Navier-Stokes equations. In The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), Wiley, 1996, pp. 175-194. · Zbl 0891.76051
[10] M. Feistauer, J. Felcman and V. Dolejší: Numerical simulation of compresssible viscous flow through cascades of profiles. ZAMM 76 (1996), 297-300. · Zbl 0925.76443
[11] M. Feistauer, J. Felcman and M. Lukáčová: On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Numer. Methods Partial Differential Equations 13 (1997), 163-190. <a href=”http://dx.doi.org/10.1002/(SICI)1098-2426(199703)13:23.0.CO;2-N” target=”_blank”>DOI 10.1002/(SICI)1098-2426(199703)13:23.0.CO;2-N |
[12] M. Feistauer, J. Felcman and M. Lukáčová: Combined finite elements-finite volume solution of compressible flow. J. Comput. Appl. Math. 63 (1995), 179-199. · Zbl 0852.76040
[13] M. Feistauer, J. Felcman, M. Lukáčová and G. Warnecke: Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. · Zbl 0960.65098
[14] M. Feistauer, J. Felcman, M. Rokyta and Z. Vlášek: Finite element solution of flow problems with trailing conditions. J. Comput. Appl. Math. 44 (1992), 131-145. · Zbl 0766.76049
[15] M. Feistauer, J. Slavík and P. Stupka: On the convergence of the combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes. Numer. Methods Partial Differential Equations
[16] J. Felcman: Finite volume solution of the inviscid compressible fluid flow. ZAMM 72 (1992), 513-516. · Zbl 0825.76666
[17] J. Felcman and V. Dolejší: Adaptive methods for the solution of the euler equations in elements of the blade bachines. ZAMM 76 (1996), 301-304. · Zbl 0925.76438
[18] J. Felcman, V. Dolejší and M. Feistauer: Adaptive finite volume method for the numerical solution of the compressible euler equations. In Computational Fluid Dynamics ’94, J. Périaux S. Wagner, E. H. Hirschel and R. Piva (eds.), John Wiley and Sons, Stuttgart, 1994, pp. 894-901.
[19] J. Felcman and G. Warnecke: Adaptive computational methods for gas flow. In Proceedings of the Prague Mathematical Conference, Prague, ICARIS, 1996, pp. 99-104. · Zbl 0963.76555
[20] J. Fořt, M. Huněk, K. Kozel and M. Vavřincová: Numerical simulation of steady and unsteady flows through plane cascades. In Numerical Modeling in Continuum Mechanics II, R. Ranacher M. Feistauer and K. Kozel (eds.), Faculty of Mathematics and Physics, Charles Univ., Prague, 1995, pp. 95-102. · Zbl 0854.76058
[21] R. Glowinski, J. L. Lions and R. Trémolières: Analyse numérique des inéquations variationnelles. Dunod, Paris, 1976.
[22] T. Ikeda: Maximum principle in finite element models for convection-diffusion phenomena. In Mathematics Studies 76, Lecture Notes in Numerical and Applied Analysis Vol. 4, North-Holland, Amsterdam-New York-Oxford, 1983. · Zbl 0508.65049
[23] C. Johnson: Finite element methods for convection-diffusion problems. In Computing Methods in Engineering and Applied Sciences V, Glowinski R. and Lions J. L. (eds.), North-Holland, Amsterdam, 1981.
[24] D. Kröner: Numerical Schemes for Conservation Laws. Wiley-Teubner, Stuttgart, 1997. · Zbl 0872.76001
[25] A. Kufner, O. John and S. Fučík: Function Spaces. Academia, Prague, 1977.
[26] K. W. Morton: Numerical Solution of Convection-Diffusion Problems. Chapman & Hall, London, 1996. · Zbl 0861.65070
[27] K. Ohmori and T. Ushijima: A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations. RAIRO, Anal. Numér. 18 (1984), 309-322. · Zbl 0586.65080
[28] H.-G. Roos, M. Stynes and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, 24, Springer-Verlag, Berlin, 1996. · Zbl 0844.65075
[29] F. Schieweck and L. Tobiska: A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. \(M^2AN\) 23 (1989), 627-647. · Zbl 0681.76032
[30] G. Strang: Variational crimes in the finite element method. In The Mathematical Foundations of the Finite Element Method, A. K. Aziz (ed.), Academic Press, New York, 1972, pp. 689-710. · Zbl 0264.65068
[31] R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford, 1979. · Zbl 0454.35073
[32] L. Tobiska: Full and weighted upwind finite element methods. In Splines in Numerical Analysis Mathematical Research Volume, Volume 32, J. W. Schmidt, H. Spath (eds.), Akademie-Verlag, Berlin, 1989. · Zbl 0685.65074
[33] G. Vijayasundaram: Transonic flow simulation using an upstream centered scheme of Godunov in finite elements. J. Comp. Phys. 63 (1986), 416-433. · Zbl 0592.76081
[34] G. Zhou: A local \({L}^2\)-error analysis of the streamline diffusion method for nonstationary convection-diffusion systems. \(M^2AN \) 29 (1995), 577-603. · Zbl 0839.65100
[35] G. Zhou and R. Rannacher: Pointwise superconvergence of streamline diffusion finite-element method. Numer. Methods Partial Differential Equations 12 (1996), 123-145. <a href=”http://dx.doi.org/10.1002/(SICI)1098-2426(199601)12:13.0.CO;2-U” target=”_blank”>DOI 10.1002/(SICI)1098-2426(199601)12:13.0.CO;2-U |
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