A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

Wang, Song; Li, Zi-Cai

doi:10.1090/S0025-5718-03-01516-3

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation
HTML articles powered by AMS MathViewer

by Song Wang and Zi-Cai Li PDF

Math. Comp. 72 (2003), 1689-1709 Request permission

Abstract:

In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter $\varepsilon$. The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by $\displaystyle C h\sqrt {\left | \frac {\ln \varepsilon }{\ln h}\right |}$ with $C$ independent of the mesh parameter $h$, the diffusion coefficient $\varepsilon$ and the exact solution of the problem.

References

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Lutz Angermann, Error estimates for the finite-element solution of an elliptic singularly perturbed problem, IMA J. Numer. Anal. 15 (1995), no. 2, 161–196. MR 1323737, DOI 10.1093/imanum/15.2.161
Wilhelm Heinrichs, Spectral multigrid methods for domain decomposition problems using patching techniques, Appl. Math. Comput. 59 (1993), no. 2-3, 165–176. MR 1253159, DOI 10.1016/0096-3003(93)90087-U
Randolph E. Bank, Josef F. Bürgler, Wolfgang Fichtner, and R. Kent Smith, Some upwinding techniques for finite element approximations of convection-diffusion equations, Numer. Math. 58 (1990), no. 2, 185–202. MR 1069278, DOI 10.1007/BF01385618
I. Christie, D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz, Finite element methods for second order differential equations with significant first derivatives, Internat. J. Numer. Methods Engrg. 10 (1976), no. 6, 1389–1396. MR 445844, DOI 10.1002/nme.1620100617
M. Dobrowolski and H.-G. Roos, A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes, Z. Anal. Anwendungen 16 (1997), no. 4, 1001–1012. MR 1615644, DOI 10.4171/ZAA/801
Miloslav Feistauer, Jiří Felcman, and Mária Lukáčová-Medviďová, Combined finite element–finite volume solution of compressible flow, J. Comput. Appl. Math. 63 (1995), no. 1-3, 179–199. International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994). MR 1365559, DOI 10.1016/0377-0427(95)00051-8
Miloslav Feistauer, Jiří Felcman, and Mária Lukáčová-Medviďová, On the convergence of a combined finite volume–finite element method for nonlinear convection-diffusion problems, Numer. Methods Partial Differential Equations 13 (1997), no. 2, 163–190. MR 1436613, DOI 10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-N
Miloslav Feistauer, Jan Slavík, and Petr Stupka, On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes, Numer. Methods Partial Differential Equations 15 (1999), no. 2, 215–235. MR 1674294, DOI 10.1002/(SICI)1098-2426(199903)15:2<215::AID-NUM6>3.0.CO;2-1
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
J.C. Heinrich, P.S. Huyakorn, A.R. Mitchell, and O.C. Zienkiewicz, “An upwind finite element scheme for two-dimensional convective transport equations", Internat. J. Num. Meth. Engng. 11 (1977) 131-143.
T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979) AMD, vol. 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979, pp. 19–35. MR 571681
C. Johnson “Streamline diffusion methods for problems in fluids” in Finite elements in fluids, vol. VI, R.H. Gallagher et al. (eds.) John Wiley and Sons, London (1986) 251-261.
J. J. H. Miller and S. Wang, A new nonconforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem, IMA J. Numer. Anal. 14 (1994), no. 2, 257–276. MR 1268995, DOI 10.1093/imanum/14.2.257
John J. H. Miller and Song Wang, An exponentially fitted finite volume method for the numerical solution of $2$D unsteady incompressible flow problems, J. Comput. Phys. 115 (1994), no. 1, 56–64. MR 1300331, DOI 10.1006/jcph.1994.1178
J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Error estimates in the maximum norm for linear problems in one and two dimensions. MR 1439750, DOI 10.1142/2933
H.-G. Roos, M. Stynes, and L. Tobiska, Numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. MR 1477665, DOI 10.1007/978-3-662-03206-0
Hans-Görg Roos, Dirk Adam, and Andreas Felgenhauer, A novel nonconforming uniformly convergent finite element method in two dimensions, J. Math. Anal. Appl. 201 (1996), no. 3, 715–755. MR 1400562, DOI 10.1006/jmaa.1996.0283
Mirko Sardella, On a coupled finite element–finite volume method for convection-diffusion problems, IMA J. Numer. Anal. 20 (2000), no. 2, 281–301. MR 1752266, DOI 10.1093/imanum/20.2.281
D. Scharfetter, H.K. Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Trans. Elec. Dev., ED-16, 64–77 (1969) 64–77.
T. Skalický, H.-G. Roos, D., “Galerkin/least-squared finite element method for convection-diffusion problems on Gartland meshes”, Report MATH-NM-12-98, Technical University of Dresden (1998).
Martin Stynes and Eugene O’Riordan, A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214 (1997), no. 1, 36–54. MR 1645503, DOI 10.1006/jmaa.1997.5581
Song Wang, A novel exponentially fitted triangular finite element method for an advection-diffusion problem with boundary layers, J. Comput. Phys. 134 (1997), no. 2, 253–260. MR 1458829, DOI 10.1006/jcph.1997.5691
I. E. Anufriev and L. V. Petukhov, Application of a stable boundary analogue of the collocation method for the approximation of the solution of some problems in mechanics, Prikl. Mat. Mekh. 62 (1998), no. 4, 633–642 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 62 (1998), no. 4, 589–597. MR 1680363, DOI 10.1016/S0021-8928(98)00075-6