Single-grid spectral collocation for the Navier-Stokes equations

The aim of the paper is to study a collocation spectral method to approximate the Navier-Stokes equations: only one grid is used, which is built from the nodes of a Gauss-Lobatto quadrature formula, either of Legendre or of Chebyshev type. The convergence is proven for the Stokes problem provided with inhomogeneous Dirichlet conditions, then thoroughly analyzed for the Navier-Stokes equations. The practical implementation algorithm is presented, together with numerical results

Viscous instability of a compressible round jet

The compressible linear stability equations are derived from the Navier Stokes equations in cylindrical polar coordinates. Numerical solutions for locally parallel flow are found using a direct matrix method. Discretization with compact finite difference is found to have better convergence properties than a Chebyshev spectral method for a round jet test case. The method is validated against previous results and convergence is tested for a range of jet profiles. Finally and en method is used to determine the dominant frequency of a Mach 0.9 jet

Finite length effects in Taylor-Couette flow

Axisymmetric numerical solutions of the unsteady Navier-Stokes equations for flow between concentric rotating cyclinders of finite length are obtained by a spectral collocation method. These representative results pertain to two-cell/one-cell exchange process, and are compared with recent experiments

Fast Ewald summation for free-space Stokes potentials

We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e. sums involving a large number of free space Green's functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems, with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid. Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of O(N log N) for problems with N sources and targets. Comparison is made with a fast multipole method (FMM) to show that the performance of the new method is competitive.Comment: 35 pages, 15 figure

Adaptive mesh strategies for the spectral element method

An adaptive spectral method was developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the 1-D viscous Burger equation and the 2-D Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities, and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility, and general capabilities for high order spectral methods

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

Nonconforming mortar element methods: Application to spectral discretizations

Spectral element methods are p-type weighted residual techniques for partial differential equations that combine the generality of finite element methods with the accuracy of spectral methods. Presented here is a new nonconforming discretization which greatly improves the flexibility of the spectral element approach as regards automatic mesh generation and non-propagating local mesh refinement. The method is based on the introduction of an auxiliary mortar trace space, and constitutes a new approach to discretization-driven domain decomposition characterized by a clean decoupling of the local, structure-preserving residual evaluations and the transmission of boundary and continuity conditions. The flexibility of the mortar method is illustrated by several nonconforming adaptive Navier-Stokes calculations in complex geometry

Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{{ \mathcal{N}}} - P_{{ \mathcal{N}}-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P 1−P 1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipe