Regularized Adaptive Methods for Classes of Nonlinear Partial Differential Equations

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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.