Adaptive FEM for Controlling Pointwise Errors and Level Sets

This proposal involves the development and analysis of adaptive finite element methods (AFEM) for the reliable and efficient computation of pointwise quantities and level sets in elliptic problems. The proposer has previously proved a posteriori error estimates for controlling pointwise and local pointwise gradient errors in second-order scalar elliptic PDE using AFEM. This project consists of three main components. First, we will further develop fundamental theory for a posteriori control of pointwise and local errors in scalar elliptic problems. The second component is the extension of the proposer's theory for a posteriori control of pointwise and local pointwise gradient errors in elliptic scalar problems to elliptic systems arising in important applications, in particular the stationary Stokes system and the equations of linear elasticity. The latter application is of particular practical interest in engineering computations as it seeks to address the pointwise control of errors in the computation of stresses. The third main goal is to develop adaptive algorithms for computations in which the desired output is a subset of the computational domain (e.g., a level set) instead of some norm of the error. In the context of static problems, examples include computing the location where the maximum temperature is taken on in a body at thermal equilibrium, or where the stresses in a body under a particular load exceed a given threshold. Answering these questions rigorously via AFEM involves control of pointwise errors, and answering them with maximum efficiency necessitates controlling errors locally. Thus we will develop an algorithm based on our previously established local pointwise a posteriori estimates.