Robust Solvers for Coupled Problems with Applications to Electromagnetism and Poromechanics

该项目的目标是为描述耦合物理问题的大型方程组开发、实现和研究鲁棒且高效的计算（偏微分方程）求解器。特别是，我们的目标是研究在障碍物周围电磁现象的计算中的应用，以及孔隙力学的应用，例如多孔介质中地下水流动的研究。设计这些求解器代表了计算数学中一类重要的具有挑战性的问题，因为这些耦合系统通常描述跨越不同时间和空间尺度的复杂多物理场现象。目前，最有效和健壮的求解器是针对单一物理问题开发的，而我们开发的每个工具都将强烈考虑固有耦合的重要性。该研究将为电磁学和孔隙弹性学提供新的计算范式，这两者在物理和工程中都有重要的应用，如聚变能应用、页岩气开采、二氧化碳固化和心肌行为等。最后，该项目支持了一名研究生。通过与该领域的调查人员和其他专家的培训和合作，他们将参与更广泛的科学计算和工程研究社区。上述每一种应用都对应于一个离散的偏微分方程（PDEs）耦合系统。由于这些模型所描述的多物理场和多尺度现象的复杂性，由于模拟复杂物理场所需的计算时间在许多情况下主要是求解表示离散偏微分方程的大规模线性方程组，因此有效且鲁棒的非线性和线性求解器是必不可少的组成部分。因此，本研究的重点是开发、分析和实现耦合PDE系统的有效迭代方法和前置条件。更确切地说，这可以通过两种可能的方法来实现。当应用保持结构的离散化时，PDE模型的属性被转移到离散模型中，用于导出精确的块分解，我们将设计新的预调节器，而不近似Schur补。对于一般数值格式，将通过推广不定耦合系统的多网格理论框架来发展整体多网格方法。最后，通过应用这些新的迭代解算器并研究它们与精确离散化方案的相互作用，研究人员将为麦克斯韦方程组和描述孔隙力学的方程组等系统创建健壮而有效的数值模拟。更一般地说，这里开发的新求解器将提供如何在代数求解器的设计中使用分析工具的见解，并将考虑为一般耦合偏微分方程的鲁棒求解器的设计提供新的理论基础。

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.