Least-Squares Finite Element Methods for PDEs in Nonsmooth Domains

Chad Westphal
Wabash College

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 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 solid and fluid
mechanics.