Jakub Both: Tuesday 25 February at 15:00, A415 Inria Paris.
Coupled flow and mechanical deformation of porous media has been of increased interest in the recent past with applications ranging from geotechnical to biomedical engineering. With increased model complexity a high demand in numerical solvers arises. In this context, physically-based iterative splitting solvers, sequentially solving the physical subproblems, have been widely popular due to their simple implementation and the possibility of reusing existing solver technologies. For unconditional stability however suitable and model-dependent stabilization is typically required. In the previous literature, the main motivation for specific choices has mostly been based on physical intuition. In this talk, a systematic development of such solvers is presented based on mathematical justification.
A gradient flow framework is presented for the modeling, analysis, and development of numerical solvers for coupled processes in poroelastic media. Various existing poroelasticity models fall into the framework, e.g., the linear Biot equations but also extensions involving viscoelastic, thermal, and/or nonlinear material laws. Besides of enabling abstract tools for the well-posedness analysis, the approach naturally leads to robust physically-based iterative splitting solvers. Gradient flow formulations are naturally discretized in time using a series of (convex) optimization problems. In the spirit of splitting solvers, we propose applying the fundamental alternating minimization for a systematic and robust decoupling of the physical subproblems. By this we re-discover popular solvers as the undrained and fixed-stress splits for the linear Biot equations, and we also provide novel iterative splittings for more advanced models. A priori convergence is established in a unified fashion utilizing abstract convergence theory for alternating minimization. This is joint work with Kundan Kumar, Jan M. Nordbotten, and Florin A. Radu (all UiB).