Marc Gerritsma

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.

Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.

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

Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution

In this paper we apply the recently developed mimetic discretization method to the mixed formulation of the Stokes problem in terms of vorticity, velocity and pressure. The mimetic discretization presented in this paper and in Kreeft et al. [51] is a higher-order method for curvilinear quadrilaterals and hexahedrals. Fundamental is the underlying structure of oriented geometric objects, the relation between these objects through the boundary operator and how this defines the exterior derivative, representing the grad, curl and div, through the generalized Stokes theorem. The mimetic method presented here uses the language of differential k-forms with k-cochains as their discrete counterpart, and the relations between them in terms of the mimetic operators: reduction, reconstruction and projection. The reconstruction consists of the recently developed mimetic spectral interpolation functions. The most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level. As a result operators like gradient, curl and divergence are discretized exactly. For Stokes flow, this implies a pointwise divergence-free solution. This is confirmed using a set of test cases on both Cartesian and curvilinear meshes. It will be shown that the method converges optimally for all admissible boundary conditions.

Compatible Spectral Approximations for the Velocity-Pressure-Stress Formulation of the Stokes Problem

The numerical approximation of the mixed velocity-pressure-stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces, a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. The discrete problem is analyzed and error estimates are derived. A comparison between the mixed approach and the standard velocity-pressure formulation is made in terms of the condition number of the resulting systems.

Edge Functions for Spectral Element Methods

It is common practice in finite element methods to expand the unknowns in nodal functions. The discretization of the gradient, curl and divergence operators requires H 1, H(curl) and H(div) function spaces and their discrete representation. Especially in mixed formulations this involved quite some mathematical machinery which can be avoided once we recognize that not all unknowns are associated with point-values. In this short paper higher order basis functions will be presented which have the property that conservation laws become independent of the basis functions. The basis functions proposed in this paper yield a discrete representation of grad, curl and div which are exact and completely determined by the topology of the grid. The discretization of these vector operators is invariant under general C 1 transformations.

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.

The use of Chebyshev Polynomials in the space-time least-squares spectral element method

Chebyshev polynomials of the first kind 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. An edge detection method is employed to determine the position of the discontinuity. Piecewise reconstruction of the numerical solution retrieves a monotone solution. Numerical results will be given in which the capabilities of the space-time formulation to capture discontinuities will be demonstrated.

Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms

This paper introduces the basic concepts for physics-compatible discretization techniques. The paper gives a clear distinction between vectors and forms. Based on the difference between forms and pseudo-forms and the @?-operator which switches between the two, a dual grid description and a single grid description are presented. The dual grid method resembles a staggered finite volume method, whereas the single grid approach shows a strong resemblance with a finite element method. Both approaches are compared for the Poisson equation for volume forms. By defining a suitably weighted inner product for 1-forms this approach can readily be applied to anisotropic diffusion models for volume forms.

Higher-Order Gauss---Lobatto Integration for Non-Linear Hyperbolic Equations

Least-squares spectral elements are capable of solving non-linear hyperbolic equations, in which discontinuities develop in finite time. In recent publications [De Maerschalck, B., 2003, http://www.aero.lr.tudelft.nl/∼bart; De Maerschalck, B., and Gerritsma, M. I., 2003, AIAA; De Maerschalck, B., and Gerritsma, M. I., 2005, Num. Algorithms, 38(1–3); 173–196], it was noted that the ability to obtain the correct solution depends on the type of linearization, Picard’s method or Newton linearization. In addition, severe under-relaxation was necessary to reach a converged solution. In this paper the use of higher-order Gauss–Lobatto integration will be addressed. When a sufficiently fine GL-grid is used to approximate the integrals involved, the discrepancies between the various linearization methods are considerably reduced and under-relaxation is no longer necessary