A New Method of Stabilization for Singular Perturbation Problems with Spectral Methods

2006

J Eng Math

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A least-squares spectral collocation scheme for the incompressible Navier-Stokes equations is proposed. Grid refinement is performed by means of adaptive triangular elements. On each triangle the Fekete nodes are employed for the collocation of the differential equation. On the element interfaces continuity of the functions is enforced in the least-squares sense. Equal-order Dubiner polynomials are used to obtain a stable spectral scheme. The collocation conditions and the interface conditions lead to an overdetermined system that can be solved efficiently by least-squares. The solution technique only involves symmetric positive-definite linear systems. The approach is first applied to the Poisson equation and then extended to singular perturbation problems where least-squares have a stabilizing effect. By adaptivity, a suitable decomposition of the domain is found where the boundary layer is well resolved. Finally, the method is successfully applied to the regularized driven-cavity flow problem. Numerical simulations confirm the high accuracy of the proposed spectral least-squares scheme.

Section: Introductionsupporting

confidence: 53%

“…It is still possible to vary the polynomial order from element to element. -Stability for singular perturbation problems [6,7] and the Stokes or Navier-Stokes equations [34,35,38,40,41]. -Adaptivity and a posteriori error estimators.…”

Section: Introductionmentioning

confidence: 99%

An adaptive least-squares spectral collocation method with triangular elements for the incompressible Navier–Stokes equations

2006

J Eng Math

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“…where @T denotes the boundary of T. We apply the standard Chebyshev collocation scheme to the exact solution u(x; y) = xy(e x+y e): (1) This function obviously ful lls the boundary condition. As we see the high spectral accuracy can also be reached on the triangle T. We have the best approximation of the solution at P(0,1) as the collocation points cluster there.…”

mentioning

confidence: 99%

Spectral Schemes on Triangular Elements

Journal of Computational Physics

Loch

2001

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The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The mapping between square and triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be applied. Numerical experiments demonstrate the expected high spectral accuracy. Further, it is shown that nite di erence preconditioning can be successfully applied in order to construct an e cient iterative solver. Then the convection-di usion equation is considered. Here nite di erence preconditioning with central di erences does not overcome instability. However, applying the rst order upstream scheme, we obtain a stable method. Finally, a domain decomposition technique is applied to the patching of rectangular and triangular elements.

“…• equal order interpolation polynomials can be employed • improved stability properties for singular perturbation problems [8,13,15] and the Navier-Stokes equations [9,16,[23][24][25] • good performance in combination with the overlapping Schwarz method • direct or iterative solvers for positive definite systems (e.g., Cholesky or conjugate gradient methods) can be used • implementation and parallelization is straightforward.…”

Section: Introductionmentioning

confidence: 99%

Least-squares spectral collocation with the overlapping Schwarz method for the incompressible Navier–Stokes equations

2006

Numer Algor

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A least-squares spectral collocation scheme is combined with the overlapping Schwarz method. The methods are succesfully applied to the incompressible Navier-Stokes equations. The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The overlapping Schwarz method is used for the iterative solution. For parallel implementation the subproblems are solved in a checkerboard manner. Our approach is successfully applied to the lid-driven cavity flow problem. Only a few Schwarz iterations are necessary in each time step. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.