Orthogonal spline collocation methods for biharmonic problems

Lou, Zhou-Ming; Bialecki, Bernard; Fairweather, Graeme

doi:10.1007/s002110050368

Cited by 22 publications

(30 citation statements)

References 22 publications

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“…The quadrature Galerkin problem of the present article is studied in [6] for existence, uniqueness, and convergence of a solution. The optimal order error estimates in Sobolev norms are obtained under a sufficient regularity assumption using the fact that the quadrature Galerkin problem is equivalent to an orthogonal spline collocation scheme, which is related to that in [9]. Multilevel preconditioners are developed and analyzed in [7], which are similar to those in this work, for the solution of a Hermite orthogonal spline collocation discretization of a second-order boundary value problem on a rectangle with a non-self-adjoint, indefinite operator.…”

Section: Introductionmentioning

confidence: 99%

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Aitbayev

2006

Numer. Methods Partial Differential Eq.

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Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the BognerFox-Schmit rectangular element and the product two-point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners.

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Section: Introductionmentioning

confidence: 99%

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Aitbayev

2006

Numer. Methods Partial Differential Eq.

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“…To rewrite (3.2) as a SchrOdinger system, we introduce the functions [105] on matrix decomposition methods for the biharmonic Dirichlet problem, it is believed that it is not possible to formulate a standard ADI method for the case of clamped boundary conditions (1.22) because in this problem there are boundary conditions on only one ofthe unknowns, Ul, a known complication in the biharmonic Dirichlet problem. However, a capacitance matrix method, which was used effectively in [105] for the solution of the biharmonic Dirichlet problem, can be employed to solve the Crank-Nicolson scheme efficiently for the special case of constant a and piecewise Hermite bicubics on an N x N uniform partition of O.…”

Section: Two Dimensional Problemsmentioning

confidence: 99%

Numerical Methods for Schrödinger-Type Problems

Fairweather¹,

Khebchareon²

2002

Applied Optimization

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Several physical phenomena are modeled by initial-boundary value problems which can be formulated as SchrOdinger -type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrodinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrodinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving SchrOdinger-type systems are also outlined.Keywords: Schrodinger systems, linear and nonlinear Schrodinger equations, parabolic wave equation, vibrating beams and plates, fourth order parabolic equations, finite difference methods, finite element Galerkin methods, orthogonal spline collocation methods, spectral methods IntroductionWe consider the numerical solution of problems that can be written as initialboundary value problems (IBVPs) for real systems of partial differential equations of the form

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“…We prove existence and uniqueness of the solution to (4.2)-(4.3) following the proof of Theorem 5.1 in [15]. To this end, we require an additional notation and two lemmas.…”

Section: Biharmonic Spectral Collocation Problemmentioning

confidence: 99%

“…Further references to the application of spectral methods to fourth order problems can be found in [7] and [3]. The formulation of the biharmonic Legendre spectral collocation problem in this paper and the method of its solution are similar to those developed in [15] and [5] for orthogonal spline collocation with piecewise Hermite bicubics.…”

Section: Introductionmentioning

confidence: 97%

A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem

Bialecki¹,

Karageorghis

2000

ESAIM: M2AN

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Abstract.A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a preconditioned conjugate gradient method. The total cost of the algorithm is O (N 3 ). Numerical results demonstrate the spectral convergence of the method.Mathematics Subject Classification. 65N35, 65N22.

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Orthogonal spline collocation methods for biharmonic problems

Cited by 22 publications

References 22 publications

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Numerical Methods for Schrödinger-Type Problems

A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem

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