Hyesuk Lee

A defect-correction method for the incompressible Navier-Stokes equations

A defect-correction method for the incompressible Navier-Stokes equation with a high Reynolds number is considered. In the defect step, the artificial viscosity parameter @s is added to the Reynolds number as a stability factor, and the residual is taken care of in the correction step. H^1 and L^2 error estimations are derived for the one-step defect-correction method, and the results of some numerical experiments are presented. These results show that, for the driven cavity, two defect-correction steps antidiffuse the artificial viscosity approximation nearly optimally. This combination gives on a very coarse mesh, results indistinguishable from a benchmark, very fine mesh calculation.

An Optimization-Based Domain Decomposition Method for the Navier--Stokes Equations

An optimization-based, nonoverlapping domain decomposition method for the solution of the Navier--Stokes equations is presented. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the Navier--Stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces. We show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined. Finite element approximations of the solutions of the optimality system are examined. The domain decomposition method is also reformulated as a nonlinear least-squares problem and the results of some numerical experiments are given.

A two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge. In this paper a two-parameter defect-correction method for viscoelastic fluid flow is presented and analyzed. In the defect step the Weissenberg number is artificially reduced to solve a stable nonlinear problem. The approximation is then improved in the correction step using a linearized correction iteration. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.

A Multigrid Method for Viscoelastic Fluid Flow

We study a multigrid finite element method for a viscoelastic fluid flow obeying an Oldroyd-B--type constitutive law. The multigrid method is a time-saving method in which the full nonlinear system is solved on a coarse grid, and subsequent approximations are generated on a succession of refined grids by solving a linearized problem. We show that the linearized problem has an approximate solution and present an error bound for a two-grid method. We also numerically demonstrate that the multigrid method is significantly more efficient than the standard one-grid finite element method.

Approximation of the Stokes---Darcy System by Optimization

A solution algorithm for the linear/nonlinear Stokes–Darcy coupled problem is proposed and investigated. The coupled system is formulated as a constrained optimal control problem, where a flow balance is forced across the interface, inflow, and outflow boundaries by minimizing a suitably defined functional. Optimization is achieved by exploiting a Neumann type boundary condition imposed on each subproblem as a control. A numerical algorithm is presented for a least squares functional whose solution yields a minimizer of the constrained optimization problem. Numerical experiments are provided to validate accuracy and efficiency of the algorithm.

Numerical Approximation of a Quasi-Newtonian Stokes Flow Problem with Defective Boundary Conditions

In this article we study the numerical approximation of a quasi-Newtonian Stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries. A variational formulation of the problem, using Lagrange multipliers to enforce the stated flow rates, is given. The existence and the uniqueness to the continuous, and discrete, variational formulations of the solution are shown. An error analysis for the numerical approximation is also given. Numerical computations are included which demonstrate the approximation scheme studied.

On Error Analysis for the 3D Navier-Stokes Equations in Velocity-Vorticity-Helicity Form

We present a rigorous numerical analysis and computational tests for the Galerkin finite element discretization of the velocity-vorticity-helicity formulation of the equilibrium Navier-Stokes equations (NSEs). This formulation, recently derived by the authors, is the first NSE formulation that directly solves for helicity and the first velocity-vorticity formulation to naturally enforce incompressibility of the vorticity, and preliminary computations confirm its potential. We present a numerical scheme; prove stability, existence of solutions, uniqueness under a small data condition, and convergence; and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem.

A Two-Level Discretization Method for the Smagorinsky Model

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed and analyzed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We demonstrate numerically that, for an appropriate choice of grids, the two-level algorithm is significantly more efficient than the standard one-level algorithm. We also provide a rigorous numerical analysis of the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (

Analysis and approximation of the Cross model for quasi-Newtonian flows with defective boundary conditions

\delta

Two-level finite element discretization of viscoelastic fluid flow

). Our analysis provides a mathematical answer to the large eddy simulation mystery of finding the scaling between the filter radius and numerical mesh-size.