PSEUDOSPECTRAL LEAST SQUARES METHOD FOR STOKES-DARCY EQUATIONS

We investigate the first order system least squares Legendre and Chebyshev pseudospectral method for coupled Stokes-Darcy equations. A least squares functional is defined by summing up the weighted L-2-norm of residuals of the first order system for coupled Stokes-Darcy equations and that of Beavers-Joseph-Saffman interface conditions. Continuous and discrete homogeneous functionals are shown to be equivalent to a combination of weighted H(div) and H-1-norms for Stokes and Darcy equations. The spectral convergence for the Legendre and Chebyshev methods is derived. Some numerical experiments are demonstrated to validate our analysisopen0

A stabilized multiple time step method for coupled Stokes-Darcy flows and transport model

A stabilized finite element algorithm with different time steps on different physical variables for the coupled Stokes-Darcy flows system with the solution transport is studied. The viscosity in the model is assumed to depend on the concentration. The nonconforming piecewise linear Crouzeix-Raviart element and piecewise constant are used to approximate velocity and pressure in the coupled Stokes-Darcy flows system, and conforming piecewise linear finite element is used to approximate concentration in the transport system. The time derivatives are discretized with different step sizes for the partial differential equations in these two systems. The existence and uniqueness of the approximate solution are unconditionally satisfied. A priori error estimates are established, which also provides a guidance on the ratio of time step sizes with respect to the ratio of the physical parameters. Numerical examples are presented to verify the theoretical results

Reduced-Dimensional Models of Porous-Medium Convection.

We study the problem of convection in a fluid-saturated porous medium in two di- mensions. Using resolved direct numerical simulations, we show that in the turbulent regime, the width of the smallest periodic box capable of sustaining a convective cell with the “correct” large-aspect ratio vertical heat transport closely matches the average width of the minimal flow unit, i.e., the convective cell naturally developing in a large box. This suggests that the minimal flow unit is indeed the smallest autonomous dy- namical unit of the flow. Next, we develop a Galerkin spectral method using an adapted basis derived from upper-bound theory and compare its performance with that of the Fourier-Galerkin method. We show that the adapted method is superior at severe truncations, but not discernibly advantageous asymptotically.PhDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/97981/1/navid_1.pd