Linear iterative schemes for doubly degenerate parabolic equations

TL;DR

This paper introduces a robust linear iterative scheme based on the L-scheme for solving degenerate parabolic equations, ensuring convergence without regularization and demonstrating effectiveness through numerical validation.

Contribution

It develops a convergent linear iterative method for degenerate parabolic equations that avoids regularization and works under mild time step restrictions.

Findings

The scheme converges rigorously under mild time step restrictions.

Numerical results confirm the theoretical convergence and stability.

The method compares favorably with existing schemes like Newton and Picard.

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 scheme, which results in a stable and locally mass-conservative scheme. At the same time, at each time step one has to solve a non-linear algebraic system, for which linear iterations are needed. Finding robust and convergent ones is particularly challenging here, since both slow and fast diffusion cases are allowed. Commonly used schemes, like Newton and Picard iterations, 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 a linear iterative scheme which builds on the $L$-scheme, and does not employ any regularisation. We prove its rigourous convergence, which is obtained for mild restrictions on the time step. Finally, we give numerical results confirming the theoretical ones, and compare the behaviour of the scheme with other schemes.