Robust, locally efficient, and guaranteed a posteriori estimates for nonlinear elliptic/parabolic problems

Koondanibha Mitra
Eindhoven University of Technology
Tuesday 5th December 2023
Aula Saleri
We firstly consider strongly monotone and Lipschitz-continuous nonlinear elliptic problems. We apply a finite element discretization in conjunction with an iterative linearization such as the fixed-point (L-)scheme or the Newton scheme. In this setting, we derive a posteriori error estimates that are robust with respect to the ratio of the continuity over monotonicity constants in the dual energy norm invoked by the linearization iterations. This is linked to an orthogonal decomposition of the total error into a linearization error component and a discretization error component, which can be further used to adaptively stop the linearization iterations for efficient error balancing. The applications cover diverse physical phenomena such as flow through porous media, mixed dimensional flow problems, mean curvature and biological flow processes. Numerical experiments for the time-discrete Richards equation illustrate the theoretical results. We further generalize our results to the Richards equation which is a nonlinear advection-reaction-diffusion (parabolic) equation exhibiting both parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies. Reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation are derived by introducing a novel degeneracy estimator, time-integrated norms, and using the maximum principle. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for realistic cases. It is shown that the estimators correctly identify the errors up to a factor of the order of unity. Contatti: Department of Excellence initiative: This initiative is part of the activities of the "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.