CSD Seminar Series: Koondanibha Mitra

Koondi completed Bachelor/Masters from IIT Kharagpur (India) in 2015 majoring in Mechanical Engineering. He completed his Ph.D. with the highest distinction in 2019, jointly from TU Eindhoven (The Netherlands) and Hasselt University (Belgium), being supervised by Prof. Sorin Pop. Then he did successive post-docs at TU Dortmund, INRIA Paris, and Radboud University Nijmegen. Currently, he is an independent researcher in Hasselt University after receiving the Junior Postdoctoral Fellowship from the Flanders Research Foundation (FWO). His research interests are applied and numerical analysis of nonlinear degenerate PDEs, especially resulting from porous media flow problems.

Title of talk: Reliable, efficient, and robust a posteriori estimates for nonlinear elliptic/parabolic problems: applications in porous media flow

Abstract:  

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 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, mean curvature flow, and biological 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 a realistic case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.