Problems in mathematical foundations of adaptive finite element methods

The goal of this project is the development and analysis of adaptive nite element methods (AFEM) for various elliptic and parabolic partial dierential 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. These include development and analysis of adaptive finite element method for problems posed on surfaces for which only incomplete information is available, parabolic surface FEM, and surface and Hodge- Laplace eigenvalue problems. In addition, we will continue our investigation of ne properties of nite element methods by proving Lp stability of FEM on highly graded meshes and proving L1 a posteriori bounds for elliptic interface problems.