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Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations

  • Jakub Wiktor Both
  • Kundan Kumar
  • Jan Martin Nordbotten
  • Iuliu Sorin Pop
  • Florin Adrian Radu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

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).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jakub Wiktor Both
    • 1
    • 2
  • Kundan Kumar
    • 3
  • Jan Martin Nordbotten
    • 3
    • 4
  • Iuliu Sorin Pop
    • 1
    • 2
  • Florin Adrian Radu
    • 3
  1. 1.University of BergenDepartment of MathematicsBergenNorway
  2. 2.Hasselt UniversityFaculty of SciencesDiepenbeekBelgium
  3. 3.University of BergenDepartment of MathematicsBergenNorway
  4. 4.Princeton UniversityDepartment of Civil and Environmental EngineeringPrincetonUSA

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