The main topics of the proposed research are (1) the development and analysis of mesh-independent methods for computation of steady state solutions of time-dependent partial differential equations, especially those with nonsmooth nonlinearities, and (2) fast algorithms for continuation of the solutions of compact fixed point problems with respect to a parameter. The investigate will apply his results to chemical engineering, chemistry, environmental science, electrical engineering, nuclear engineering, and aeronautical engineering. Both phases of the project address a weakness of a conventional implementation of a globalized inexact Newton method, namely the inability to select among multiple solutions. Continuation methods use additional information, such as a physical parameter in the equation or the fact that the nonlinear equation is the steady state equation for a time-dependent problem, to guide the solver, and both improve the solver's robustness and increase the likelihood that the important solution or solutions will be found. The proposed research will analyze the robustness of these methods as computational grids are refined, how they perform if the nonlinearity is not differentiable, and how one can design fast algorithms for the corrector iteration which best exploit the functional analytic properties of the combination of the problem and the continuation method.
Nonlinearity is common across all of science and engineering. Therefore, algorithms for solving nonlinear equations are fundamental components of many simulators and engineering design tools. The investigator will study solution methods for equations that depend on physical parameters, such as the voltage across a semiconductor device, the load on a mechanical structure, or the temperature of a chemical reaction. As such parameters change, the nature of the solution can vary significantly, and the performance of the simulator can be affected as well. The PI's research is directed at improving the performance of the algorithms and making the solvers more robust. His success will have an impact on the performance of simulator and design tools, and thereby accelerate the design cycle. The PI's and his students, through collaborations with several national laboratories and companies, will apply these results to problems in nano-scale electronics, chemical engineering, and aerospace.
