Mathematical Sciences: Numerical Approximation of Strongly Nonlinear and Constrained Problems

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This research will consider the numerical analysis of strongly nonlinear problems including free boundary problems, constrained problems such as Stokes flow, and the Hamilton-Jacobi equations. Linearization techniques based on nonlinear Chernoff formulae will be investigated, and nonlinear algorithms for both parabolic and eliptic problems will be analyzed with particular emphasis on accuracy in nonenergy spaces and splitting algorithms. The impact of local mesh refinements in reducing computational labor for a given accuracy will be considered. Work analyzing the pointwise accuracy of mixed finite element methods will be continued. This work will combine and interrelate theory and numerical methods. It should increase understanding of both the numerical techniques and the motivating physical situations.