Solution of degenerate parabolic variational inequalities with convection
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, p. 417-431

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.

DOI : https://doi.org/10.1051/m2an:2003035
Classification:  65M25,  65M12
Keywords: Richard's equation, convection-diffusion, parabolic variational inequalities
@article{M2AN_2003__37_3_417_0,
     author = {Kacur, Jozef and Keer, Roger Van},
     title = {Solution of degenerate parabolic variational inequalities with convection},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {3},
     year = {2003},
     pages = {417-431},
     doi = {10.1051/m2an:2003035},
     zbl = {1033.65049},
     mrnumber = {1994310},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_3_417_0}
}
Kacur, Jozef; Keer, Roger Van. Solution of degenerate parabolic variational inequalities with convection. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, pp. 417-431. doi : 10.1051/m2an:2003035. http://www.numdam.org/item/M2AN_2003__37_3_417_0/

[1] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341. | Zbl 0497.35049

[2] H.W. Alt, S. Luckhaus and A. Visintin, On the nonstationary flow through porous media. Ann. Math. Pura Appl. CXXXVI (1984) 303-316. | Zbl 0552.76075

[3] J. Babušikova, Application of relaxation scheme to degenerate variational inequalities. Appl. Math. 46 (2001) 419-439. | Zbl 1061.49004

[4] J.W. Barrett and P. Knabner, 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. | Zbl 0904.76039

[5] J.W. Barrett and P. Knabner, 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. | Zbl 0911.65078

[6] J. Bear, Dynamics of Fluid in Porous Media. Elsevier, New York (1972).

[7] R. Bermejo, Analysis of an algorithm for the Galerkin-characteristics method. Numer. Math. 60 (1991) 163-194. | Zbl 0723.65073

[8] R. Bermejo, A Galerkin-characteristics algorithm for transport-diffusion equation. SIAM J. Numer. Anal. 32 (1995) 425-455. | Zbl 0854.65083

[9] C.N. Dawson, C.J. Van Duijn and M.F. Wheeler, Characteristic-Galerkin methods for contaminant transport with non-equilibrium adsorption kinetics. SIAM J. Numer. Anal. 31 (1994) 982-999. | Zbl 0808.76046

[10] R Douglas and T.F. Russel, 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. | Zbl 0492.65051

[11] R.E. Ewing and H. Wang, Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis. Comput. Mech. 12 (1993) 97-121. | Zbl 0774.76058

[12] R. Eymard, M. Gutnic and D. Hilhorst, The finite volume method for Richards equation. Comput. Geosci. 3 (1999) 259-294. | Zbl 0953.76060

[13] P. Frolkovic, Flux-based method of characteristics for contaminant transport in flowing groundwater. Computing and Visualization in Science 5 (2002) 73-83. | Zbl 1052.76578

[14] R. Glowinski, J.-L. Lions and R. Tremolieres, Numerical analysis of variational inequalities, Vol. 8. North-Holland Publishing Company, Stud. Math. Appl. (1981). | MR 635927 | Zbl 0463.65046

[15] A. Handlovicova, Solution of Stefan problems by fully discrete linear schemes. Acta Math. Univ. Comenianae (N.S.) 67 (1998) 351-372. | Zbl 0930.65108

[16] H. Holden, K.H. Karlsen and K.-A. Lie, Operator splitting methods for degenerate convection-diffusion equations II: numerical examples with emphasis on reservoir simulation and sedimentation. Comput. Geosci. 4 (2000) 287-323. | Zbl 1049.35113

[17] W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Modelling Numer. Anal. 29 (1995) 605-627. | Numdam | Zbl 0837.65103

[18] J. Kačur, Solution of some free boundary problems by relaxation schemes. SIAM J. Numer. Anal. 36 (1999) 290-316. | Zbl 0924.65090

[19] J. Kačur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal. 19 (1999) 119-154. | Zbl 0946.65145

[20] J. Kačur, Solution of degenerate convection-diffusion problems by the method of characteristics. SIAM J. Numer. Anal. 39 (2001) 858-879. | Zbl 1011.65064

[21] J. Kačur and S. Luckhaus, Approximation of degenerate parabolic systems by nondegenerate alliptic and parabolic systems. Appl. Numer. Math. 25 (1997) 1-21. | Zbl 0894.65043

[22] J. Kačur and R. Van Keer, Solution of contaminant transport with adsorption in porous media by the method of characteristics. ESAIM: M2AN 35 (2001) 981-1006. | Numdam | Zbl 0995.76070

[23] A. Kufner, O. John and S. Fučík, Function spaces. Academia, Prague (1977). | MR 482102 | Zbl 0364.46022

[24] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol. XX. Dunod, Gauthier-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[25] K. Mikula, Numerical solution of nonlinear diffusion with finite extinction phenomena. Acta Math. Univ. Comenian. (N.S.) 2 (1995) 223-292. | Zbl 0852.35080

[26] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Academia, Prague (1967). | MR 227584

[27] F. Otto, L1 - contraction and uniqueness for quasilinear elliptic - parabolic equations. C. R. Acad. Sci Paris Sér. I Math. 321 (1995) 105-110. | Zbl 0845.35056

[28] P. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38 (1982) 309-332. | Zbl 0505.76100

[29] X. Shi, H. Wang and R.E. Ewing, An ellam scheme for multidimensional advection-reaction equations and its optimal-order error estimate. SIAM J. Numer. Anal. 38 (2001) 1846-1885. | Zbl 1006.76074