A Numerical Approach to a Nonlinear and Degenerate Parabolic Problem by Regularization Scheme

Abstract

In this work we propose a numerical scheme for a nonlinear and degenerate parabolic problem having application in petroleum reservoir and groundwater aquifer simulation. The degeneracy of the equation includes both locally fast and slow diffusion (i.e. the diffusion coefficients may explode or vanish in some point). The main difficulty is that the true solution is typically lacking in regularity. Our numerical approach includes a regularization step and a standard discretization procedure by means of C0-piecewise linear finite elements in space and backward-differences in time. Within this frame work, we analyze the accuracy of the scheme by using an integral test function and obtain several error estimates in suitable norms.

Share and Cite:

Cao, H. (2014) A Numerical Approach to a Nonlinear and Degenerate Parabolic Problem by Regularization Scheme. Journal of Applied Mathematics and Physics, 2, 88-93. doi: 10.4236/jamp.2014.25012.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Alt, H.W. (1985) Nonsteady Fluid Flow through Porous Media. Applications and Theory, 3, 222-228.
[2] Damlamina, A. (1977) Some Result in the Multiphase Stefan Problem. Communications in Partial Differential Equations, 2, 1017-1044. http://dx.doi.org/10.1080/03605307708820053
[3] van Duijn C.J. and Peletier L.A. (1982) Nonstationary Filtration in Partially Saturated Porous Media. Archive for Rational Mechanics and Analysis, 78, 173-198. http://dx.doi.org/10.1007/BF00250838
[4] Nochetto, R.H. and Verdi, C. (1988) Approximation of Degenerate Parabolic Problems Using Numerical Intergration. SIAM Journal on Numerical Analysis, 25, 784-814. http://dx.doi.org/10.1137/0725046
[5] Alt, H.W. and Luckhaus, S. (1983) Quasilinear Elliptic-Parabolic Differential Equations. Mathematische Zeitschrift, 183, 311-341. http://dx.doi.org/10.1007/BF01176474
[6] Ladyzenskaya O., Solonnikov V. and Ural’ceva N. (1968) Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs.
[7] Nochetto R.H. (1987) Error Estimates for Multidimensional Singular Parabolic Problems. Japan Journal of Industrial and Applied Mathematics, 4, 111-138. http://dx.doi.org/10.1007/BF03167758
[8] Pop, I.S. (2002) Error Estimates for a Time Discretization Method for Richards’ Equation. Computers & Geosciences, 6, 141-160. http://dx.doi.org/10.1023/A:1019936917350
[9] Radu, F., Pop, I.S. and Knabner, P. (2004) Oder of Convergence Estimates for an Euler Implicit Mixed Finite Element Discretization of Richards’ Equation. SIAM Journal on Numerical Analysis, 22, 1452-1478. http://dx.doi.org/10.1137/S0036142902405229
[10] Pop, I.S. and Yong, W.A. (1997) A maximum Principle Based Numerical Approach to Porous Medium Equation. Proceedings of the 14th Conference on Scientific Computing, 207-218.
[11] Arbogast, T., Wheeler, M.F. and Zhang, N.Y. (1996) A Nonlinear Mixed Finite Element Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media. SIAM Journal on Numerical Analysis, 33, 1669-1687. http://dx.doi.org/10.1137/S0036142994266728
[12] Jager, W. and Kacur, J. (1995) Solution of Doublely Nonlinear and Degenerate Parabolic Problems by Relaxation Schemes. Mathematical Modelling and Numerical Analysis, 29, 605-627.
[13] Adams, R.A. and Fournier, J.J.F. (2003) Sobolev Spaces. 2nd Edition, Academic Press, New York.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.