Regularized Adaptive Methods for Classes of Nonlinear Partial Differential Equations

非线性扩散方程经常出现在热传导、地下水流动和多孔介质流动等物理过程的模拟中。这些方程出现在描述地下污染物分布的与环境有关的建模问题中。在这些非线性问题中，扩散系数依赖于解，可能还取决于它的坐标；因此，模型的数值解不能用直接方法确定。一般来说，只有通过解一系列更简单的近似问题，并对解进行不断的改进，才能找到解决方案。然而，如果以标准的方式进行，则该格式的收敛可能会失败，并且当前的数值模拟可能会受到这些标准迭代技术的失败的限制。这项工作的重点是开发数学上严格的稳定和收敛的迭代数值算法来高效地求解非线性扩散方程，解决现实物理模型的科学计算中的一个实质性问题。该项目的技术目标是开发高效和健壮的非线性扩散方程类的模拟技术。众所周知，非线性扩散问题的有限元解在渐近区域具有良好的逼近性质，但计算这些离散解的合理方法尚未得到发展。正则化的自适应方法将在自适应有限元方法的框架内发展，其(1)迭代收敛到离散解；(2)随着网格的选择性加密，离散解收敛到偏微分方程解。这项工作的目的之一是制定与误差指标相一致的诱导离散问题正则化的指导原则，以确定网格细化。正则化和误差指标的组合应该既允许计算离散解，又从理论上保证收敛到正确的解。以完善的数学理论为后盾的计算方法将被开发用于典型的模型问题类别，所开发的方法将被扩展到更大的、计算要求高的模拟，例如模拟注入地球地下的二氧化碳的分布，作为一种潜在的长期存储手段。预计在该项目过程中取得的技术进步将促进高效和准确的数值模拟工具的实现，从而能够对具有真实物理属性的问题进行实际建模。

Nonlinear diffusion equations appear often in simulations of physical processes such as heat conduction, groundwater flow, and flow in porous media. Such equations appear in environmentally relevant modeling problems describing the distribution of subsurface contaminants. In these nonlinear problems, the diffusion coefficient is dependent on the solution and possibly its coordinates; as such, a numerical solution to the model cannot be determined by direct means. Generally, solutions can only be found by solving a sequence of simpler approximate problems, and making successive improvements to the solution. Convergence of the scheme may fail however if this is carried out in a standard way, and current numerical simulations can be limited by the failure of these standard iterative techniques. The focus of this work is on the mathematically rigorous development of stable and convergent iterative numerical algorithms to efficiently solve nonlinear diffusion equations, addressing a substantial problem in scientific computing for realistic physical modeling.The technical goal of this project is to develop efficient and robust simulation technology for classes of nonlinear diffusion equations. Finite element solutions for nonlinear diffusion problems are known to have good approximation properties in the asymptotic regime, but a sound methodology to compute those discrete solutions has yet to be developed. Regularized adaptive methods will be developed within the framework of adaptive finite element methods, for which (1) the iterates converge to discrete solutions; and (2) the discrete solutions converge to the solution of the partial differential equation, as the mesh is selectively refined. One of the aims of this work is to develop guiding principles for the regularization of the induced discrete problems in concert with error indicators to determine the mesh refinement. The combination of the regularization and error indicators should both allow the computation of a discrete solution, and guarantee the convergence to a correct solution from a theoretical standpoint. Computational methods backed up by sound mathematical theory will be developed for representative classes of model problems, and the developed methods will be extended to larger, computationally-demanding simulations, for example to model the distribution of C02 injected into the earth's subsurface as a potential means for long-term storage. It is expected that the technical advances made in the course of this project will advance the realization of efficient and accurate numerical simulation tools that allow the practical modeling of problems with realistic physical attributes.