Adaptive FEM for Elliptic and Parabolic Problems

---...., /" Adaptive FEM for elliptic and parabolic problems The goal of this project is the development and analysis of adaptive finite element methods (AFEM) for various elliptic and parabolic partial differential equations. AFEM are widely used in physical simulations involving PDE, and the proposed work will broaden the applicability and deepen our understanding of this important tool. The three main subprojects of the proposal all continue the development of themes and techniques present in the PI's previous work. The first subproject involves theoretical investigation of convergence properties of AFEM for controlling non-standard norms and related applications.. It is only in the past few years that quasi-optimality results for adaptive methods have appeared. Because of their strong reliance on Galerkin orthogonality, previous results mainly concern AFEM for global energy norms. The PI has developed techniques for similarly analyzing AFEM designed to control other norms. He will apply these techniques to prove optimality of an AFEM for controlling local energy norms and use them to analyze convergence of parallel adaptive algorithms of Bank-Holst type. In the second set of projects the PI will continue his collaboration with European colleagues to develop AFEM for efficiently controlling local and global maximum errors in several classes of parabolic problems where such error control is desirable. These efforts will result in sharp new a posteriori error estimates and adaptive finite element methods for controlling local pointwise errors in linear parabolic problems and maJdmum-norm a posteriori error analyses of semilinear parabolic problems for which such estimates are of practical interest. The third project involves development of AFEM for solving elliptic and parabolic PDE on surfaces. We will develop a surface AFEM for stationary problems which will be useful for problems in which surface and bulk effects are coupled, while our study of linear parabolic surface PDE will provide foundational knowledge about AFEM on evolving surfaces.