Michael M. J. Proot

A Least-Squares Spectral Element Formulation for the Stokes Problem

Least-squares spectral element methods seem very promising since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The new element in this work is the choice of spectral elements for the discretization of the least-squares formulation for its superior accuracy due to the high-order basis-functions. The main issue of this paper is the derivation of a least-squares spectral element formulation for the Stokes equations and the role of the boundary conditions on the coercivity relations. The numerical simulations confirm the usual exponential rate of convergence when p-refinement is applied which is typical for spectral element discretization.

Mass- and Momentum Conservation of the Least-Squares Spectral Element Method for the Stokes Problem

The opinion that least-squares methods are not useful due to their poor mass conserving property should be revised. It will be shown that least-squares spectral element methods perform poorly with respect to mass conservation, but this is compensated with a superior momentum conservation. With these new insights, one can firmly state that the least-squares spectral element method remains an interesting alternative for the commonly used Galerkin spectral element formulation

Analysis of a Discontinuous Least Squares Spectral Element Method

This paper addresses the development of a Discontinuous Spectral Least-Squares method. Based on pre-multiplication with a mesh-dependent function a discontinuous functional can be set up. Coercivity of this functional will be established. An example of the approximation to a continuous solution and a solution in which a jump is prescribed will be presented. The discontinuous least-squares method preserves symmetry and positive definiteness of the discrete system.

Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations

In this paper the extension of the Legendre least-squares spectral element formulation to Chebyshev polynomials will be explained. The new method will be applied to the incompressible Navier-Stokes equations and numerical results, obtained for the lid-driven cavity flow at Reynolds numbers varying between 1000 and 7500, will be compared with the commonly used benchmark results. The new results reveal that the least-squares spectral element formulations based on the Legendre and Chebyshev Gauss-Lobatto Lagrange interpolating polynomials are equally accurate.

Space-Time Least-Squares Spectral Elements for Convection Dominated Unsteady Flows

Legendre polynomials are employed in a space-time least-squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations. No stabilization techniques are required to render a stable, high-order-accurate scheme. In parts of the domain where the underlying exact solution is smooth, the scheme exhibits exponential convergence with polynomial enrichment, whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a Gibbs-like behavior. Numerical results will be given in which the capabilities of the space-time formulation to capture discontinuities will be demonstrated.

Parallel Implementation of a Least-Squares Spectral Element Solver for Incompressible Flow Problems

Least-squares spectral element methods (LSQSEM) are based on two important and successful numerical methods: spectral/hp element methods and least-squares finite element methods. Least-squares methods lead to symmetric and positive definite algebraic systems which circumvent the Ladyzhenskaya–Babuška–Brezzi (LBB) stability condition and consequently allow the use of equal order interpolation polynomials for all variables. In this paper, we present results obtained with a parallel implementation of the least-squares spectral element solver on a distributed memory machine (Cray T3E) and on a virtual shared memory machine (SGI Origin 3800).

A parallel least-squares spectral element solver for incompressible flow problems on unstructured grids

The parallelisation of the least-squares spectral element formulation of the Stokes problem is discussed for incompressible flow problems on unstructured grids. The method leads to a large symmetric positive definite algebraic system, that is solved iteratively by the conjugate gradient method. To improve the convergence rate, both Jacobi and Additive Schwarz preconditioners are applied. Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large-scale incompressible flows on a Cray T3E and on an SGI Origin 3800.

A parallel, state-of-the-art, least-squares spectral element solver for incompressible flow problems

The paper deals with the efficient parallelization of least-squares spectral element methods for incompressible flows. The parallelization of this sort of problems requires two different strategies. On the one hand, the spectral element discretization benefits from an element-by-element parallelization strategy. On the other hand, an efficient strategy to solve the large sparse global systems benefits from a row-wise distribution of data. This requires two different kinds of data distributions and the conversion between them is rather complicated. In the present paper, the different strategies together with its conversion are discussed. Moreover, some results obtained on a distributed memory machine (Cray T3E) and on a virtual shared memory machine (SGI Origin 3800) are presented.

Application of least-squares spectral element solver methods to incompressible flow problems

textabstractLeast-squares spectral element methods are based on two important and successful numerical methods: spectral /hp element methods and least-squares finite element methods. In this respect, least-squares spectral element methods are very powerfull since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The present paper continues the development of the least-squares spectral element methods by concentrating on the application of this method to incompressible flow problems. Therefore, the derivation of the least-squares spectral element formulation of the velocity-vorticity-pressure form of the unsteady Navier-Stokes equations plays a central role in the present paper. Moreover, the numerical simulation of the lid driven cavity problem confirms that the least-squares spectral element method produces spectrally accurate results.