Adaptive FEM for controlling pointwise errors and level sets

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Elliptic partial differential equations are ubiquitous in science and engineering applications, and their fast and accurate numerical solution remains an important area of research. Adaptive finite element algorithms for automatically constructing efficient computational grids are very popular tools for solving such equations. An adaptive finite element method (AFEM) is an iterative feedback procedure in which an initial approximate solution is computed, and information from the initial approximation is then used to construct a better approximation. What is meant here by "better approximation" depends upon the desired output from the computation. Most adaptive codes are designed to control the energy norm (root-mean-square, or average, of the first derivatives) of the error because the energy norm is closely associated with the finite element algorithm. The goal output from many computations, on the other hand, is related to other ways of measuring the error, and there is generally no guarantee that control of the energy error will lead to computationally efficient control of other measures of the error. Much recent research has thus focused on using adaptive codes to compute "quantities of interest" not related to the energy norm.This project involves the construction and analysis of adaptive algorithms for computations where the goal quantity is either related to pointwise information about the error, or is a location within the overall computational domain. Situations where such information is desirable include locating the maximum temperature in a body at thermal equilibrium and determining where the stresses in an elastic body exceed a given threshold. Level set methods in which evolving interfaces are represented as level sets of solutions to partial differential equations have also gained popularity in recent years. This project contains three main goals. First, the PI and others have previously developed several aspects of a basic theory for a posteriori estimation of pointwise errors in simple model problems. This theory will be enriched and extended. Secondly, we will investigate application of this basic theory to systems important in applications, in particular the stationary Stokes system from fluid dynamics and equations of linear elasticity. Finally, we will develop an adaptive algorithm that rigorously controls level sets of solutions to elliptic problems. The proposed research also provides for the training of a graduate student in numerical analysis and scientific computing.