September 5 – Koondi Mitra: A fast and stable linear iterative scheme for nonlinear parabolic problems

Koondi Mitra: Thursday 5 September at 15:00, A415 Inria Paris.

We consider nonlinear parabolic problems of the general form
tb(u) + ∇ · F(u) = ∇ · [D(u)∇p] + S(u),
where the variable p can be expressed as p = f(u,∂u).

Such problems find applications in nonlinear diffusion, crowd dynamics and for the main focus of this work, flow through porous media. Due to lack of time regularity of the solutions, backward Euler method is used to discretize such equations in time which gives a sequence of nonlinear elliptic problems. Linear iterative schemes such as the Newton or the Picard scheme are often used to solve the nonlinear elliptic problems. However, their stability can only be ensured under strong constraints on the time step size and provided that the problem does not become degenerate. An alternative is the L-scheme, discussed in [2, 3], which guarantees convergence of the iterations even for degenerate cases with a minor constraint in time step size. However, it is considerably slower compared to the Newton and the Picard scheme [3]. Since for nonlinear parabolic problems, we have a good initial guess to start the iterations in the form of the solution of the previous time step, we propose a modified version of the L-scheme [1] that takes this into account. It is proved that it converges linearly even for degenerate cases with a minor constraint in time step size. The linear convergence rate is propotional to an exponent of the time step size for this scheme, which in practice makes it faster than both the L-scheme and the Picard scheme, and more stable than the Newton and the Picard scheme. This is supported by numerical computations. The scheme is extended to many different problems such as the two phase flow problem and domain decomposition methods.

References
[1] Mitra, K., Pop., I.S.: A modified L-scheme to solve nonlinear diffusion problems. Computers & Mathematics with Applications, Vol. 77 (2019), pp. 1722-1738
[2] Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the Richards’equation: linearization procedure. Journal of Computational & Applied Mathematics, Vol. 168 (2004), pp. 365-373
[3] List, F., Radu, F.A.: A study on iterative methods for Richards’ equation, Computational Geo- sciences, Vol. 20 (2016), pp. 341-353

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