Robust Solvers for Coupled Problems with Applications to Electromagnetism and Poromechanics

这个项目的目标是为描述耦合物理问题的大规模方程组开发、实现和研究健壮和高效的计算(偏微分方程)解算器。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，例如研究多孔介质中的地下水流动。设计这些解算器代表了计算数学中一类重要的挑战性问题，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现象。目前，最有效和最健壮的解算器是为单物理问题开发的，而我们开发的每个工具都会强烈考虑内在耦合的重要性。这项研究将为电磁学和孔弹性提供新的计算范例，这两者在物理和工程中都有关键的应用，例如聚变能源应用、页岩气回收、二氧化碳固结和心肌行为，仅举几例。最后，该项目资助了一名研究生。通过培训和与该领域的调查人员和其他专家合作，他们将参与更广泛的科学计算和工程研究社区。上述每个应用对应于离散化的偏微分方程组(PDE)。由于这些模型所描述的多物理和多尺度现象的复杂性，由于模拟复杂物理所需的计算时间在许多情况下由求解代表离散的偏微分方程组的大规模线性方程组所支配，因此一个重要的组成部分是高效和健壮的非线性和线性求解器。因此，本研究致力于开发、分析和实现耦合偏微分方程组的高效迭代方法和预条件算子。更准确地说，这是通过两种可能的方法实现的。当应用保结构离散化时，利用带入离散模型的偏微分方程模型的性质来推导精确的块分解，并且我们将设计新的预条件算子而不需要近似Schur补。对于一般的数值格式，通过推广不定耦合系统的多重网格理论框架，发展了整体多重网格方法。最后，通过应用这些新的迭代求解器，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和描述孔隙力学的系统创建健壮而高效的数值模拟。更广泛地说，这里开发的新的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洞察力，并将考虑用于一般耦合偏微分方程组的稳健求解器设计的新的理论基础。

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.