On the approximation of the spectrum of the Stokes operator

Izvestiya, Physics of the Solid Earth, 2008

A method of solving the Stokes equation for a spherical mantle model by expansion in spherical harmonics was developed by . However, this method is applicable only if the viscosity depends solely on depth. In this case, the Stokes equation reduces to a system of independent equations for each harmonic. Given lateral variations in viscosity, the Stokes equation contains terms in the form of products of harmonics, which invalidates all advantages of harmonic expansion. Zhang and Christensen [1993] developed a perturbation method for the case when terms containing products of lateral viscosity variations are small. These terms are first calculated from the preceding iteration and are then expanded in a series of harmonic functions. As a result, equations for harmonics remain independent. An evident advantage of the spectral method is the simplicity of the technique of incorporating the self-gravitation and compressibility effects. Moreover, this method partially eliminates difficulties related to the singularities at poles. As yet, it has not been applied in practice, possibly because the equations presented in contain misprints that have not been elucidated in the literature. In the present work, a system of equations is derived for the spectral-iterative method of solving the Stokes equation and the errata present in formulas of Zhang and Christensen [1993] and significantly affecting results of calculations are analyzed. PACS numbers: 91.10.Kg

Numerical Methods for Partial Differential Equations, 2009

We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a-posteriori error bounds. The estimates are verified by numerical computations.

Numerische Mathematik, 1984

Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard FEM is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.

Computer Methods in Applied Mechanics and Engineering 80(1-3) 229-236 (1990), 1990

We propose and analyse two spectral type algorithms to approximate the Stokes problem in a square. The first one is a collocation method in which the only unknown is the stream-function, the second one is a Galerkin method involving numerical integration and using the stream-function and the vorticity as two dependent variables.

This paper describes the recently developed mixed mimetic spectral element method for the Stokes problem in the vorticity-velocity-pressure formulation. This compatible discretization method relies on the construction of a conforming discrete Hodge decomposition, that is based on a bounded projection operator that commutes with the exterior derivative. The projection operator is the composition of a reduction and a reconstruction step. The reconstruction in terms of mimetic spectral element basis-functions are tensor-based constructions and therefore hold for curvilinear quadrilateral and hexahedral meshes. For compatible discretization methods that contain a conforming discrete Hodge decomposition, we derive optimal a priori error estimates which are valid for all admissible boundary conditions on both Cartesian and curvilinear meshes. These theoretical results are confirmed by numerical experiments. These clearly show that the mimetic spectral elements outperform the commonly used...

ESAIM: Mathematical Modelling and Numerical Analysis, 2004

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

Journal of Mathematical Fluid Mechanics, 2002

We consider the Stokes system in a truncated exterior domain, with a local artificial boundary condition on the truncating sphere. The second derivatives of the velocity and the first derivatives of the pressure are estimated in the L 2 -norm, and it is shown how the constants in these estimates depend on the radius of the truncating sphere.

Journal of the Mathematical Society of Japan, 2003

This paper is concerned with the standard L p estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer. §1. Introduction. Let W H R n ðn f 2Þ be an infinite layer, i.e., W ¼fx ¼ðx 0 ; x n Þ A R n j x 0 ¼ðx 1 ; ...; x nÀ1 Þ A R nÀ1 ; 0 < x n < 1g: This paper is concerned with the resolvent problem of the Stokes operator on W with Dirichlet zero boundary condition: ðl À DÞu þ 'p ¼ f ; ' Á u ¼ 0 i n W; uj x n ¼0 ¼ uj x n ¼1 ¼ 0; 2000 Mathematics Subject Classification. 35Q30, 76D07.

Annales de l'Institut Henri Poincare (C) Non Linear Analysis

Let (SD Ω) be the Stokes operator defined in a bounded domain Ω of R 3 with Dirichlet boundary conditions. We prove that, generically with respect to the domain Ω with C 5 boundary, the spectrum of (SD Ω) satisfies a non resonant property introduced by C. Foias and J. C. Saut in [16] to linearize the Navier-Stokes system in a bounded domain Ω of R 3 with Dirichlet boundary conditions. For that purpose, we first prove that, generically with respect to the domain Ω with C 5 boundary, all the eigenvalues of (SD Ω) are simple. That answers positively a question raised by J. H. Ortega and E. Zuazua in [25, Section 6]. The proofs of these results follow a standard strategy based on a contradiction argument requiring shape differentiation. One needs to shape differentiate at least twice the initial problem in the direction of carefully chosen domain variations. The main step of the contradiction argument amounts to study the evaluation of Dirichlet-to-Neumann operators associated to these domain variations.

1981

This paper gives a theory of spectral approximation for closed operators in Banach spaces. The perturbation theory developed in this paper is applied to the study of a finite element procedure for approximating the spectral properties of a differential system modeling a fluid in a rotating basin.