Robust Solvers for Coupled Problems with Applications to Electromagnetism and Poromechanics

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The goal of this project is to develop, implement, and study robust and efficient computational (Partial Differential Equation) solvers for large-scale systems of equations that describe coupled physical problems. In particular, we aim to investigate applications in the computation of electromagnetic phenomena around obstacles, as well as poromechanic applications, such as the study of groundwater flow in porous media. Designing these solvers represents an important class of challenging problems in computational mathematics, because those coupled systems usually describe complex multiphysics phenomena across different time and spatial scales. Currently, most efficient and robust solvers are developed for single-physics problems, whereas each tool we develop will strongly consider the importance of the inherent coupling. The research will provide new computational paradigms for electromagnetics and poroelasticity, both of which have crucial applications in physics and engineering, such as fusion energy applications, shale gas recovery, carbon dioxide consolidation, and cardiac muscle behavior, to name a few. Finally, the project supports one graduate student. Through training and collaboration with investigators and other experts in the field, they will become involved in the broader research communities of scientific computing and engineering. Each of the applications described above corresponds to a discretized coupled system of partial differential equations (PDEs). Due to the complexity of the multi-physics and multi-scale phenomena described by such models, an essential component is efficient and robust nonlinear and linear solvers, due to the fact that the computational time needed to simulate complex physics is in many cases dominated by solving the large-scale linear systems of equations representing the discretized PDEs. Therefore, this research focuses on developing, analyzing, and implementing efficient iterative methods and preconditioners for coupled PDE systems. More precisely, this is achieved by two possible approaches. When structure-preserving discretizations are applied, properties of the PDE models that are carried over to the discrete model are used to derive an exact block factorization and we will design novel preconditioners without approximating the Schur complements. For general numerical schemes, monolithic multigrid methods will be developed by generalizing the multigrid theoretical framework for indefinite coupled systems. Finally, by applying these new iterative solvers and studying their interplay with accurate discretization schemes, the investigators will create robust and efficient numerical simulations for systems such as Maxwell's equations and those describing poromechanics. More generally, the new solvers developed here will provide insight on how to use analytic tools in the design of algebraic solvers and novel theoretical foundations for the design of robust solvers for general coupled PDEs will be considered.