Problems in mathematical foundations of adaptive finite element methods

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A posteriori error estimates and adaptive finite element methods (AFEM) are widely-used tools for solving partial differential equations (PDEs) arising in science and engineering applications. A posteriori estimates provide computable bounds on discretization errors, while AFEM are efficient solution techniques which accurately reflect solution properties via automatic local mesh grading. The goals of this project are to better understand the mathematical underpinnings of AFEM and to provide new a posteriori error estimates and adaptive algorithms in several specific application areas. A major part of the project is devoted to development and analysis of a posteriori error estimates and AFEM for PDEs on surfaces. Specific projects concern Eulerian formulations of parabolic PDEs on evolving surfaces, solution of elliptic PDEs on surfaces for which the only available information is a discrete approximation, and elliptic eigenvalue problems. Another emphasis is fine properties of FEM, in particular the development of a priori and a posteriori error estimates in nonstandard norms. The PI will develop new a priori error estimates in such norms on the types of highly graded meshes typically seen in practice, prove new a posteriori maximum-norm bounds for elliptic interface problems, and integrate similar error analysis into his study of surface eigenvalue problems. A wide variety of applications in science and engineering give rise to partial differential equations (PDEs) which must be solved in order to obtain accurate predictions about the physical world. PDEs are typically solved approximately on computers in modern applications, and there is a tradeoff between the quality of the approximate solution and the investment of computational resources. The PI will study the mathematical underpinnings of adaptive algorithms which automatically generate more accurate solutions while efficiently employing the computing power at hand. Part of the project is aimed at enriching mathematical understanding of existing algorithms, and part to developing new and mathematically well-justified adaptive algorithms for various applications.