Linear iterative schemes for doubly degenerate parabolic equations
This work provides a robust and convergent linear solver for challenging degenerate parabolic PDEs, addressing a bottleneck in numerical simulation of porous media flow.
The authors developed a linear iterative scheme for solving doubly degenerate parabolic equations arising in porous media flow, proving rigorous convergence under mild time-step restrictions without regularization. Numerical results confirm the theoretical findings and show competitive performance compared to Newton and Picard methods.
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.