Approximation of the Spectrum of Closed Operators: The Determination of Normal Modes of a Rotating Basin

This paper gives optimal order error estimates for the approximation of the spectral properties of a variant of the shallow water equations by a finite element procedure recently proposed by Platzman. General results on the spectral approximation of unbounded, selfadjoint operators are also given in this paper. I. Introduction. We analyze in this paper an approximation procedure to compute the normal modes of the oceans. The procedure that we analyze is a finite element method proposed by Platzman [16]. Our procedure gives a finite dimensional operator whose properties are shown to approximate those of the unbounded selfadjoint operator associated with the differential equations modeling the physical problem. The error estimates given here are optimal

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Physics-conforming constraints-oriented numerical method

Ahusborde¹,

Gruber²,

Azaïez³

et al. 2007

Phys. Rev. E

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A general high-order finite element method for solving partial differential equations, called the COnstraints Oriented Library (COOL) method, is presented. This hp approach takes into account the underlying nature of the corresponding physical problem, and thus avoids the generation of nonphysical solutions. In the COOL method, all terms in a variational form are represented by the same functional dependence and by the same regularity, thus eliminating regularity constraints imposed by standard numerical methods. External constraints, such as the incompressibility condition appearing in the Maxwell or Navier-Stokes equations, can then be satisfied identically and are eliminated algebraically. This reduces the number of variables, and leads to well-conditioned matrix problems. The consequence is that only physically relevant solutions remain. The COOL method also satisfies automatically internal constraints, such as occur for the grad(div) and curl(curl) operators, and this for any geometry. This approach can be applied to a wide range of physical problems, including fluid flows, electromagnetics, material sciences, ideal linear magnetohydrodynamic stability analysis, and Alfvén wave heating of fusion plasmas. Results obtained by applying the COOL method to the grad(div) and curl(curl) operators, the Stokes problem, and the steady and unsteady Navier-Stokes equations are presented.

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Approximation of the Spectrum of Closed Operators: The Determination of Normal Modes of a Rotating Basin

Cited by 3 publications

References 12 publications

Approximation of solution branches of nonlinear equations

Approximation of solution branches of nonlinear equations

Convergence of a finite element method for the approximation of normal modes of the oceans

Physics-conforming constraints-oriented numerical method

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