# Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations

## Abstract

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method, which results in a stable and locally mass-conservative scheme. At each time step one has to solve a non-linear algebraic system, for which one needs adequate iterative solvers. Finding robust ones is particularly challenging here, since the problems considered are double degenerate (i.e. two type of degeneracies are allowed: parabolic-elliptic and parabolic-hyperbolic).

Commonly used schemes, like Newton and Picard, are defined either for non-degenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss an iterative linearisation scheme which builds on the *L*-scheme, and does not employ any regularisation. We prove its rigorous convergence, which is obtained for Hölder type non-linearities. Finally, we present numerical results confirming the theoretical ones, and compare the behaviour of the proposed scheme with schemes based on a regularisation step.

## Notes

### Acknowledgements

The research is partially supported by the Norwegian Research Council (NFR) through the NFR-DAAD grant 255715, the VISTA project AdaSim 6367 and the project Toppforsk 250223, Lab2Field 811716, by Statoil through the Akademia Grant and by the Research Foundation-Flanders (FWO) through the Odysseus programme (project G0G1316N).

## References

- 1.H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z.
**183**, 311–341 (1983)MathSciNetCrossRefGoogle Scholar - 2.T. Arbogast, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow. J. Nonlinear Anal. Theory Methods Appl.
**19**, 1009–1031 (1992)MathSciNetCrossRefGoogle Scholar - 3.N. Bergamashi, M. Putti, Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation. Int. J. Numer. Methods Eng.
**45**, 1025–1046 (1999)MathSciNetCrossRefGoogle Scholar - 4.K. Brenner, C. Cances, Improving Newton’s method performance by parametrization: the case of the Richards equation. SIAM J. Numer. Anal.
**55**, 1760–1785 (2017)MathSciNetCrossRefGoogle Scholar - 5.F. Brezzi, M. Fortin,
*Mixed and Hybrid Finite Element Methods*(Springer, New York, 1991)CrossRefGoogle Scholar - 6.M. Celia, E. Bouloutas, R. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res.
**26**, 1483–1496 (1990)CrossRefGoogle Scholar - 7.Z. Chen, Degenerate two-phase incompressible flow. Existence, uniqueness and regularity of a weak solution. J. Differ. Equ.
**171**, 203–232 (2001)MathSciNetCrossRefGoogle Scholar - 8.L. Cherfils, C. Choquet, M.M. Diedhiou, Numerical validation of an upscaled sharp-diffuse interface model for stratified miscible flows. Math. Comput. Simul.
**137**, 246–265 (2017)MathSciNetCrossRefGoogle Scholar - 9.J. Douglas Jr., J. Roberts, Global estimates for mixed methods for second order elliptic problems. Math. Comput.
**45**, 39–52 (1985)CrossRefGoogle Scholar - 10.M.W. Farthing, F.L. Ogden, Numerical solution of Richards equation: a review of advances and challenges. Soil Sci. Soc. Am. J. (2017). https://doi.org/10.2136/sssaj2017.02.0058CrossRefGoogle Scholar
- 11.W. Jäger, J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Model. Numer. Anal.
**29**, 605–627 (1995)MathSciNetCrossRefGoogle Scholar - 12.F. Lehmann, P. Ackerer, Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transp. Porous Med.
**31**, 275–292 (1998)CrossRefGoogle Scholar - 13.F. List, F.A. Radu, A study on iterative methods for Richards’ equation. Comput. Geosci.
**20**, 341–353 (2016)MathSciNetCrossRefGoogle Scholar - 14.R.H. Nochetto, C. Verdi, Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal.
**25**, 784–814 (1988)MathSciNetCrossRefGoogle Scholar - 15.J.M. Nordbotten, M.A. Celia,
*Geological Storage of CO2. Modeling Approaches for Large-Scale Simulation*(Wiley, Hokoben, 2012)Google Scholar - 16.F. Otto,
*L*^{1}-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ.**131**, 20–38 (1996)MathSciNetCrossRefGoogle Scholar - 17.C. Paniconi, M. Putti, A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems. Water Resour. Res.
**30**, 3357–3374 (1994)CrossRefGoogle Scholar - 18.E.J. Park, Mixed finite elements for non-linear second-order elliptic problems. SIAM J. Numer. Anal.
**32**, 865–885 (1995)MathSciNetCrossRefGoogle Scholar - 19.I.S. Pop, F.A. Radu, P. Knabner, Mixed finite elements for the Richards’ equations: linearization procedure. J. Comput. Appl. Math.
**168**, 365–373 (2004)MathSciNetCrossRefGoogle Scholar - 20.F.A. Radu, I.S. Pop, P. Knabner, On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation, in
*Numerical Mathematics and Advanced Applications*ed. by A. Bermudez de Castro, D. Gomez, P. Quintela, P. Salgado (Springer, Berlin, 2006), pp. 1192–1200Google Scholar - 21.F.A. Radu, I.S. Pop, P. Knabner, Error estimates for a mixed finite element discretization of some degenerate parabolic equations. Numer. Math.
**109**, 285–311 (2008)MathSciNetCrossRefGoogle Scholar - 22.F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A convergent mass conservative numerical scheme based on mixed finite elements for two-phase flow in porous media. arHiv: 1512.08387 (2015)Google Scholar
- 23.F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math.
**289**, 134–141 (2015)MathSciNetCrossRefGoogle Scholar - 24.F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A robust, mass conservative scheme for two- phase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal.
**38**, 884–920 (2018)MathSciNetCrossRefGoogle Scholar - 25.M. Slodicka, A robust and efficient linearization scheme for doubly non-linear and degenerate parabolic problems arising in flow in porous media. SIAM J. Sci. Comput.
**23**, 1593–1614 (2002)MathSciNetCrossRefGoogle Scholar - 26.R. Temam,
*Navier-Stokes Equations: Theory and Numerical Analysis*(AMS Chelsea Publishing, Providence, 2001)CrossRefGoogle Scholar - 27.W.A. Yong, I.S. Pop, A numerical approach to porous medium equations. Preprint 95–50 (SFB 359), IWR, University of Heidelberg, 1996Google Scholar