Variational Problems in Weighted Sobolev Spaces with Applications to Computational Fluid Dynamics

We study variational problems in weighted Sobolev spaces on bounded domains with corners. The specific forms of these variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations. Specifically, we introduce new variational formulations for the Poisson and Helmholtz problems in what would be a weighted counterpart of H^2 intersection H^1_0, show the existence and uniqueness of solutions, and establish the relationship with the traditional H^1 formulations. We formulate a conforming
C^1 finite element method for solving these variational problems and establish optimal convergence rates. We apply our methods to the time-dependent Navier-Stokes equations on non-convex polygonal domains by adapting an iterative algorithm due to Liu, Liu, and Pego (Comm. Pure Appl. Math., 60 (2007)). The well-posedness of the modified algorithm follows from our analysis of the Poisson and Helmholtz problems. Numerical results for several benchmark
problems are provided.