Mathematical Sciences: Numerical Methods and Adaptivity for Nonlinear PDEs

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This project will support research in the design, implementation, and analysis of adaptive finite element methods for phase transitions and free boundary problems arising in material sciences. Numerical methods for phase transitions will be studied; the use of highly graded meshes is extremely important for phase transitions and free boundary problems which typically exhibit sharp interfaces as well as very thin transition regions. The solution, or some of its derivatives, may exhibit discontinuities or just vary very rapidly within such layers, leading to global numerical difficulties if the singularities are not properly resolved. Related sharp interface models exhibiting competition between surface tension and destabilizing mechanisms will also be studied, such as stress driven instabilities in solid crystals and solidification processes. The design of linearization techniques for strongly nonlinear partial differential equations will be studied and combined with mesh refinements. The interaction of convective fluid with free boundaries, for example that corresponding to solid coexisting with its melt and interfaces of multiphase fluids, will be analyzed numerically. Finally, mixed methods for constrained problems will be studied, with special emphasis on the Stokes flow and related free boundary problems as well as on free boundary problems for plates over obstacles. This project is concerned with the design and implementation of adaptive mesh refinements for nonlinear partial differential equations arising in thermodynamics, fluid dynamics, and elasticity. The use of highly graded meshes is important for many of these problems, which typically exhibit sharp interfaces as well as very thin transition regions. The increasing interest in such problems stems not only from their mathematical features but also their applications to phase transitions in material sciences, flame propagation, combustion theory, and crystal growth.