David Wells

Xuping Xie, David Wells, Zhu Wang et al.

Standard ROMs generally yield spurious numerical oscillations in the simulation of convection-dominated flows. Regularized ROMs use explicit ROM spatial filtering to decrease these spurious numerical oscillations. The Leray ROM is a recently introduced regularized ROM that utilizes explicit ROM spatial filtering of the convective term in the Navier-Stokes equations. This paper presents the numerical analysis of the finite element discretization of the Leray ROM. Error estimates for the ROM differential filter, which is the explicit ROM spatial filter used in the Leray ROM, are proved. These ROM filtering error estimates are then used to prove error estimates for the Leray ROM. Finally, both the ROM filtering error estimates and the Leray ROM error estimates are numerically investigated in the simulation of the two-dimensional Navier-Stokes equations with an analytic solution.

Erich L Foster, Traian Iliescu, David Wells

In this paper we proposed a two-level finite element discretization of the nonlinear stationary quasi-geostrophic equations, which model the wind driven large scale ocean circulation. Optimal error estimates for the two-level finite element discretization were derived. Numerical experiments for the two-level algorithm with the Argyris finite element were also carried out. The numerical results verified the theoretical error estimates and showed that, for the appropriate scaling between the coarse and fine mesh sizes, the two-level algorithm significantly decreases the computational time of the standard one-level algorithm.

David Wells, Jeffrey Banks

Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating accurate approximations of boundary derivatives of elliptic problems: instead of postprocessing, we describe a new continuous finite element method based on p-refinement of cells adjacent to the boundary to increase the approximation order of the derivative on the boundary itself. We prove that the order of the approximation on the p-refined cells is, in 1D, determined by the rate of convergence at the knot connecting the higher and lower order cells and that this idea can be extended, in some simple settings, to 2D problems. We verify this rate of convergence numerically with a series of experiments in both 1D and 2D.