Variational problems in weighted Sobolev spaces with applications to computational fluid dynamics.

摘要

We study variational problems in weighted Sobolev spaces on bounded domains with angular points. 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 H2 ∩ H10 , show the existence and uniqueness of solutions, and establish the relationship with the traditional H1 formulations. We formulate a conforming C 1 finite element method for solving these variational problems and prove its convergence. 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. 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.