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

Abstract

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.

Direct Minimization of the least-squares spectral element functional – Part I: Direct solver

“…If m = n, i.e. the number of unknowns in the global system equals the number of equations, the use of the weight matrix W k is inconsequential and the problem reduces to a collocation method evaluated in the GLL-points, [12][13][14], given by…”

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.…”

An adaptive spectral least-squares scheme for the Burgers equation

“…for hyperbolic problems (see [2,4,10]). and the Navier-Stokes equations (see [6,11,12,24] and Jiang et al [14][15][16][17]). -Filtering, over-collocation and adaptivity can easily be combined.…”