This research project is to develop efficient dynamic mesh-adaptation strategies for the numerical simulation of evolutionary problems arising from physical science. Mesh adaptation is an indispensable tool for use in mesh-based numerical simulation in science and engineering. Its basic idea is to put more mesh nodes in regions of large solution variation than those where the solution is smooth. In this way, fewer mesh nodes are required to reach a specified level of accuracy and thus significant economies are gained. Dynamic mesh adaptation is a type of mesh adaptation which moves mesh nodes around to follow the dynamic features of the physical problem. Its continuous nature in time makes a dynamic mesh-adaptation method the natural choice of method for use in the numerical solution of evolutionary problems.
The proposed research focuses on improving the efficiency of dynamic mesh adaptation. Topics of study include the investigation of time integration of partial differential equations on moving meshes, the development of the Schwarz waveform moving-mesh method, and the development of simple and efficient solvers for mesh equations. All studies will be targeted on evolutionary partial differential equations arising from liquid-crystal models and phase-change problems. These problems have many important industrial applications and have attracted considerable interest among scientists. Successful completion of this project will provide a powerful tool for studying the formation of solution singularities and the propagation of moving interfaces arising from these problems. Graduate students will be actively involved in the research project. Students' training will benefit from large-scale computations as well as theoretical studies.
