Christopher Coley

Christopher Coley, John A. Evans

We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar transport problem, and we prove that the method exhibits optimal convergence rates in the SUPG norm and is robust with respect to the Peclet number if the discontinuous subscale approximation space is sufficiently rich. We apply the method to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed.

Christopher Coley, Joseph Benzaken, John A. Evans

In this paper, we present a geometric multigrid methodology for the solution of matrix systems associated with isogeometric compatible discretizations of the generalized Stokes and Oseen problems. The methodology provably yields a pointwise divergence-free velocity field independent of the number of pre-smoothing steps, post-smoothing steps, grid levels, or cycles in a V-cycle implementation. The methodology relies upon Scwharz-style smoothers in conjunction with specially defined overlapping subdomains that respect the underlying topological structure of the generalized Stokes and Oseen problems. Numerical results in both two- and three-dimensions demonstrate the robustness of the methodology through the invariance of convergence rates with respect to grid resolution and flow parameters for the generalized Stokes problem as well as the generalized Oseen problem provided it is not advection-dominated.