Mathematical Sciences: Finite Element Methods for Nonlinear PDE's and Adaptivity

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9623394 Nochetto Problems governed by nonlinear partial differential equations (PDEs) which exhibit sharp or diffuse, but rapid, transitions are of primary importance in science and engineering. Liquid-solid phase transitions, multicomponent alloys, anisotropy and junctions, epitaxial growth, multiphase incompressible fluids, and advection dominated flows are relevant multiscale examples considered in this project. The design, analysis and implementation of innovative finite element methods which handle the presence of free boundaries, thin layers and constraints, and the same time cope with possible numerical pollution effects via adaptively refined meshes is central to this project. A priori stability and error analyses, together with a posteriori error control are quite delicate aspects of the proposed research that reveal the limited value of linear theory in the context of strongly nonlinear PDEs. A posteriori error estimates for degenerate parabolic PDEs must reflect the distinctive presence of free boundaries and thin transition layers. Related PDE and computational issues (implementation, recursive refinement/coarsening algorithms, multilevel solvers) will also be investigated. This project is expected to shed light on the efficient and reliable adaptive finite element computation of phase transitions in materials science, and thus have impact on high performance computing of phase transitions with surface tension effects. Understanding and computing nonlinear evolutions of corrugations observed in epitaxial growth is of fundamental importance in electronics and materials science. Novel finite element methods will be designed and used to explore a simple model for stress driven instabilities in solids. Validation of a taxis model describing coexistence and interaction of several marine species in response to foraging, predators and environmental conditions is crucial in ecology and environmental sciences. Mixed finite element me thods will be used in conjunction with exponential fitting to better exploit the structure of taxis in the underlying system of nonlinear PDEs in both extremes of advection and diffusion dominated flows.