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

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

“…In [16] and [23] we have seen, that it is better to use (4) since this leads to overdetermined linear systems of equations with lower conditions numbers. Hence, the accuracy can be increased since the round-off errors do not have such a big influence to the approximations.…”

“…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.…”

“…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. 0021 Here, we consider internal flow problems to investigate the mass and momentum conservation of the least-squares spectral collocation method (LSSCM).…”

“…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.…”