The overall effectiveness of numerical methods for partial differential equations may be severely limited by solutions that lack smoothness on a relatively small subset of the domain. Problems may have singularities induced by the geometry of the domain; convection dominated regimes may result in interior or boundary layers; discontinuous material coefficients can cause sharp gradients; or solutions may blow up at interior points when operator coefficients are singular or degenerate. This project proposes a systematic study of weighted finite element methods where standard norms and inner products are replaced with weighted norms and inner products. In the least-squares finite element setting, these weight functions serve to redefine the metric under which the error is minimized and, as such, the relative accuracy of the numerical solution can be balanced throughout the domain in an optimal way. For some problems, the right choice of weights can recover convergence where the analogous nonweighted case does not converge and, in other problems, convergence rates are enhanced by an appropriate set of weights. This project will develop robust adaptive methods for a wide class of linear and nonlinear problems, where the weights are chosen from coarse scale problems within a mesh refinement strategy.
The efficient numerical solution of partial differential equations is of great importance throughout the applied sciences. Finite element methods constitute a popular and flexible approach to solving a wide range of problems, and the development of robust, accurate, and efficient finite element algorithms is in high demand for applications including mechanics of deformable solids, fluid flow, transport, and electromagnetics. This project aims to develop a new class of adaptive finite element methods that are motivated by the success of weighted-norm least squares methods and adaptive mesh refinement algorithms. Robust adaptive algorithms allow for better numerical simulations by focusing computational resources on the most challenging aspects of the problem, increasing overall accuracy and decreasing computational time. In models of glacier flow, for example, regions of the ice near the surface and the ground require more resolution than the majority of the interior ice, and accurate models must be able to allocate the computational work in an optimal way. Successful results from this project will enhance the current understanding of how adaptive algorithms can be designed to evolve optimally from coarse scale approximations.
