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。这种方程出现在描述地下污染物分布的环境相关建模问题中。在这些非线性问题中，扩散系数依赖于解并可能依赖于它的坐标；因此，该模型的数值解不能用直接方法确定。一般来说，只能通过解决一系列更简单的近似问题，并对解决方案进行连续改进来找到解决方案。然而，如果以标准的方式进行，该方案的收敛可能会失败，并且当前的数值模拟可能受到这些标准迭代技术失败的限制。这项工作的重点是在数学上严格发展稳定和收敛的迭代数值算法，以有效地解决非线性扩散方程，解决现实物理建模科学计算中的实质性问题。本课题的技术目标是开发高效、鲁棒的非线性扩散方程仿真技术。众所周知，非线性扩散问题的有限元解在渐近范围内具有良好的近似性质，但尚未开发出一种可靠的方法来计算这些离散解。正则化自适应方法将在自适应有限元方法的框架内发展，其中(1)迭代收敛于离散解；(2)由于网格有选择地细化，离散解收敛于偏微分方程的解。这项工作的目的之一是为与误差指标一致的诱导离散问题的正则化制定指导原则，以确定网格精细化。正则化和误差指标的结合应该既允许离散解的计算，又保证从理论的角度收敛到正确的解。有可靠的数学理论支持的计算方法将用于有代表性的模型问题，并且已开发的方法将扩展到更大的、对计算有要求的模拟，例如，模拟注入地球地下的二氧化碳的分布，作为长期储存的潜在手段。预计在这个项目过程中取得的技术进步将推动实现高效和准确的数值模拟工具，使具有现实物理属性的问题的实际建模成为可能。

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.