# Solution of degenerate parabolic variational inequalities with convection

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 3, page 417-431
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topKacur, Jozef, and Van Keer, Roger. "Solution of degenerate parabolic variational inequalities with convection." ESAIM: Mathematical Modelling and Numerical Analysis 37.3 (2010): 417-431. <http://eudml.org/doc/194171>.

@article{Kacur2010,

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.
},

author = {Kacur, Jozef, Van Keer, Roger},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Richard's equation; convection-diffusion; parabolic variational
inequalities.; convection-diffusion variational inequality; parabolic variational inequalities; method of characteristics; convergence},

language = {eng},

month = {3},

number = {3},

pages = {417-431},

publisher = {EDP Sciences},

title = {Solution of degenerate parabolic variational inequalities with convection},

url = {http://eudml.org/doc/194171},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Kacur, Jozef

AU - Van Keer, Roger

TI - Solution of degenerate parabolic variational inequalities with convection

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 3

SP - 417

EP - 431

AB -
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.

LA - eng

KW - Richard's equation; convection-diffusion; parabolic variational
inequalities.; convection-diffusion variational inequality; parabolic variational inequalities; method of characteristics; convergence

UR - http://eudml.org/doc/194171

ER -

## References

top- H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z.183 (1983) 311–341. Zbl0497.35049
- H.W. Alt, S. Luckhaus and A. Visintin, On the nonstationary flow through porous media. Ann. Math. Pura Appl. CXXXVI (1984) 303–316. Zbl0552.76075
- J. Babušikova, Application of relaxation scheme to degenerate variational inequalities. Appl. Math.46 (2001) 419–439. Zbl1061.49004
- 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. Zbl0904.76039
- 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. Zbl0911.65078
- J. Bear, Dynamics of Fluid in Porous Media. Elsevier, New York (1972). Zbl1191.76001
- R. Bermejo, Analysis of an algorithm for the Galerkin-characteristics method. Numer. Math.60 (1991) 163–194. Zbl0723.65073
- R. Bermejo, A Galerkin-characteristics algorithm for transport-diffusion equation. SIAM J. Numer. Anal.32 (1995) 425–455. Zbl0854.65083
- 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. Zbl0808.76046
- 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. Zbl0492.65051
- 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. Zbl0774.76058
- R. Eymard, M. Gutnic and D. Hilhorst, The finite volume method for Richards equation. Comput. Geosci.3 (1999) 259–294. Zbl0953.76060
- P. Frolkovic, Flux-based method of characteristics for contaminant transport in flowing groundwater. Computing and Visualization in Science5 (2002) 73–83. Zbl1052.76578
- R. Glowinski, J.-L. Lions and R. Tremolieres, Numerical analysis of variational inequalities, Vol. 8. North-Holland Publishing Company, Stud. Math. Appl. (1981). Zbl0463.65046
- A. Handlovicova, Solution of Stefan problems by fully discrete linear schemes. Acta Math. Univ. Comenianae (N.S.)67 (1998) 351–372. Zbl0930.65108
- 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. Zbl1049.35113
- W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Modelling Numer. Anal.29 (1995) 605–627. Zbl0837.65103
- J. Kačur, Solution of some free boundary problems by relaxation schemes. SIAM J. Numer. Anal.36 (1999) 290–316. Zbl0924.65090
- J. Kačur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal.19 (1999) 119–154. Zbl0946.65145
- J. Kačur, Solution of degenerate convection-diffusion problems by the method of characteristics. SIAM J. Numer. Anal.39 (2001) 858–879. Zbl1011.65064
- J. Kačur and S. Luckhaus, Approximation of degenerate parabolic systems by nondegenerate alliptic and parabolic systems. Appl. Numer. Math.25 (1997) 1–21.
- J. Kačur and R. van Keer, Solution of contaminant transport with adsorption in porous media by the method of characteristics. ESAIM: M2AN35 (2001) 981–1006. Zbl0995.76070
- A. Kufner, O. John and S. Fučík, Function spaces. Academia, Prague (1977). Zbl0364.46022
- J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol. XX. Dunod, Gauthier-Villars, Paris (1969). Zbl0189.40603
- K. Mikula, 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).
- F. Otto, L1 – contraction and uniqueness for quasilinear elliptic – parabolic equations. C. R. Acad. Sci Paris Sér. I Math.321 (1995) 105–110.
- P. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math.38 (1982) 309–332. Zbl0505.76100
- 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. Zbl1006.76074

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.