Mathematical Sciences: Numerical Approximation of Strongly Nonlinear and Constrained Problems

本研究将考虑强非线性问题的数值分析，包括自由边界问题，约束问题，如斯托克斯流，和汉密尔顿-雅可比方程。将研究基于非线性Chernoff公式的线性化技术，并分析抛物线和椭圆型问题的非线性算法，特别强调在非能量空间和分裂算法中的精度。在给定精度下，局部网格细化对减少计算劳动的影响将被考虑。分析混合有限元法逐点精度的工作将继续进行。这项工作将理论和数值方法结合起来。它应该增加对数值技术和激励物理情况的理解。

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.