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

Abstract

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.

A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier–Stokes equations

“…Spectral least-squares for the Navier-Stokes equations were first presented by Pontaza and Reddy in [24][25][26], followed by Gerritsma and Proot in [29]. Heinrichs investigated least-squares spectral collocation schemes in [14][15][16] that lead to symmetric and positive definite algebraic systems which circumvent the LBB stability condition. Since we here work on least-squares spectral collocation schemes we want to summarize some advantages of this approach: equal order interpolation polynomials can be employed; it is possible to vary the polynomial order from element to element; improved stability properties for small perturbation parameters in singular perturbation problems [8,13] and Stokes or Navier-Stokes equations [14][15][16][27][28][29]; good performance in combination with domain decomposition techniques; direct and efficient iterative solvers for positive definite systems can be used; implementation is straightforward.…”

Conservation of mass and momentum of the least-squares spectral collocation scheme for the Stokes problem

“…To avoid these difficulties the original domain can be decomposed into several sub domains and least-squares techniques can be applied, see, e.g. [11][12][13][14][15][16]22,23,[28][29][30][31][32][33][34]. Least-squares techniques for such problems offer theoretical and numerical advantages over the classical Galerkin-type methods which must fulfill the well-posedness (or stability) criterion, the so called LBB condition (see [5]).…”

“…Spectral least-squares for the Navier-Stokes equations were first presented by Pontaza and Reddy in [28][29][30], followed by Gerritsma and Proot in [34]. Heinrichs investigated least-squares spectral collocation schemes in [13][14][15][16] that lead to symmetric and positive definite algebraic systems which circumvent the LBB stability condition. Furthermore, Heinrichs and Kattelans presented in [16,23] least-squares spectral collocation schemes where they improved the conditions numbers of the algebraic systems, considered different types of decompositions of the domain and different interface conditions between the elements for the Stokes and Navier-Stokes equations.…”

“…equal order interpolation polynomials can be employed it is possible to vary the polynomial order from element to element improved stability properties for small perturbation parameters in singular perturbation problems [11] and Stokes or Navier-Stokes equations [13][14][15][16]23,31,32,34] good performance in combination with domain decomposition techniques direct and efficient iterative solvers for positive definite systems can be used implementation is straightforward.…”

High-order accurate collocations and least squares method for solving the Navier-Stokes equations

“…Its capabilities are enhanced when it is combined with the least squares method, which, due to the minimization of the residual functional, pro duces a more accurate numerical solution than that constructed by the collocation method alone. The capabilities of versions of the collocations and least squares (CLS) method as applied to various problems were examined, for example, in [2][3][4][5][6][7].…”