Mathematics Colloquium: Least-Squares Finite Element Methods for Partial Differential Equations in Nonsmooth Domains
4:10 p.m. Neill Hall 5W
Abstract: Partial differential equations posed on domains with polygonal /polyhedral corners may have reduced regularity, and numerical methods often suffer as a consequence. Global reduction of discretization convergence rates or even convergence to the wrong solution can occur. In this talk we'll discuss a specific methodology to combat this within the least-squares finite element context. By replacing the standard Sobolev norms in a least-squares functional with appropriately weighted norms, we are able to eliminate the global “pollution effect” caused by the nonsmooth solution and recover better (often optimal) rates of convergence in both weighted and nonweighted norms. Examples are given in both div\curl and H(div) settings as well as extensions to applications in Non-Newtonian fluid mechanics.