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Solution of degenerate parabolic variational inequalities withconvection

Published online by Cambridge University Press:  15 April 2004

Jozef Kacur  and
Roger Van Keer
Jozef Kacur
Affiliation:
Faculty of Mathematics, Physics and Informatics, Comenius University Mlynská dolina, 84248 Bratislava, Slovakia. kacur@fmph.uniba.sk.
Roger Van Keer
Affiliation:
Ghent University, Department of Mathematical Analysis, Galglaan 2, 9000 Gent, Belgium. rvk@cage.rug.ac.be.
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Abstract

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Keywords

Richard's equationconvection-diffusionparabolic variational inequalities.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 3 , May 2003 , pp. 417 - 431
Copyright
© EDP Sciences, SMAI, 2003

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References

Alt, H.W. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311–341.
H.W. Alt, S. Luckhaus and A. Visintin, On the nonstationary flow through porous media. Ann. Math. Pura Appl. CXXXVI (1984) 303–316.
Babušikova, J., Application of relaxation scheme to degenerate variational inequalities. Appl. Math. 46 (2001) 419–439. CrossRef
Barrett, J.W. and Knabner, P., Finite element approximation of transport of reactive solutes in porous media. II: Error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34 (1997) 455–479. CrossRef
Barrett, J.W. and Knabner, P., An improved error bound for a Lagrange-Galerkin method for contaminant transport with non-lipschitzian adsorption kinetics. SIAM J. Numer. Anal. 35 (1998) 1862–1882. CrossRef
J. Bear, Dynamics of Fluid in Porous Media. Elsevier, New York (1972).
Bermejo, R., Analysis of an algorithm for the Galerkin-characteristics method. Numer. Math. 60 (1991) 163–194. CrossRef
Bermejo, R., Galerkin-characteristics, A algorithm for transport-diffusion equation. SIAM J. Numer. Anal. 32 (1995) 425–455. CrossRef
Dawson, C.N., Van Duijn, C.J. and Wheeler, M.F., Characteristic-Galerkin methods for contaminant transport with non-equilibrium adsorption kinetics. SIAM J. Numer. Anal. 31 (1994) 982–999. CrossRef
Douglas, R and Russel, T.F., Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19 (1982) 871–885. CrossRef
Ewing, R.E. and Wang, H., Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis. Comput. Mech. 12 (1993) 97–121. CrossRef
Eymard, R., Gutnic, M. and Hilhorst, D., The finite volume method for Richards equation. Comput. Geosci. 3 (1999) 259–294. CrossRef
Frolkovic, P., Flux-based method of characteristics for contaminant transport in flowing groundwater. Computing and Visualization in Science 5 (2002) 73–83.
R. Glowinski, J.-L. Lions and R. Tremolieres, Numerical analysis of variational inequalities, Vol. 8. North-Holland Publishing Company, Stud. Math. Appl. (1981).
Handlovicova, A., Solution of Stefan problems by fully discrete linear schemes. Acta Math. Univ. Comenianae (N.S.) 67 (1998) 351–372.
Holden, H., Karlsen, K.H. and Lie, K.-A., Operator splitting methods for degenerate convection-diffusion equations II: numerical examples with emphasis on reservoir simulation and sedimentation. Comput. Geosci. 4 (2000) 287–323. CrossRef
Jäger, W. and Kačur, J., Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Modelling Numer. Anal. 29 (1995) 605–627. CrossRef
Kačur, J., Solution of some free boundary problems by relaxation schemes. SIAM J. Numer. Anal. 36 (1999) 290–316. CrossRef
Kačur, J., Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal. 19 (1999) 119–154. CrossRef
Kačur, J., Solution of degenerate convection-diffusion problems by the method of characteristics. SIAM J. Numer. Anal. 39 (2001) 858–879. CrossRef
Kačur, J. and Luckhaus, S., Approximation of degenerate parabolic systems by nondegenerate alliptic and parabolic systems. Appl. Numer. Math. 25 (1997) 1–21.
Kačur, J. and van Keer, R., Solution of contaminant transport with adsorption in porous media by the method of characteristics. ESAIM: M2AN 35 (2001) 981–1006. CrossRef
A. Kufner, O. John and S. Fučík, Function spaces. Academia, Prague (1977).
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol. XX. Dunod, Gauthier-Villars, Paris (1969).
Mikula, K., Numerical solution of nonlinear diffusion with finite extinction phenomena. Acta Math. Univ. Comenian. (N.S.) 2 (1995) 223–292.
J. Nečas, Les méthodes directes en théorie des équations elliptiques. Academia, Prague (1967).
Otto, F., L1 – contraction and uniqueness for quasilinear elliptic – parabolic equations. C. R. Acad. Sci Paris Sér. I Math. 321 (1995) 105–110.
Pironneau, P., On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38 (1982) 309–332. CrossRef
Shi, X., Wang, H. and Ewing, R.E., An ellam scheme for multidimensional advection-reaction equations and its optimal-order error estimate. SIAM J. Numer. Anal. 38 (2001) 1846–1885.