Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Gia, Quôc Thông Lê; Sloan, Ian H.; Tran, Thanh

doi:10.1090/s0025-5718-08-02150-9

Cited by 14 publications

(18 citation statements)

References 29 publications

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“…The numbers, reported in Table 7, clearly show that the bound for λ min (P) is better than that of λ min (A). Details of this calculation have been reported in [7].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In [7] we study the use of additive Schwarz preconditioners for elliptic partial differential equations on the sphere, and prove a bound for the condition number. A similar approach will be used in the present paper for the interpolation problem.…”

Section: Introductionmentioning

confidence: 99%

“…A similar approach will be used in the present paper for the interpolation problem. However, a more complicated analysis for the present case has to be carried out since the setting of the interpolation problem is in Sobolev spaces of any real order, instead of the simple space H 1 (S n ) studied in [7].…”

Section: Introductionmentioning

confidence: 99%

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An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

Gia

Tran

2010

Journal of Complexity

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a b s t r a c tThe problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.Crown

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“…The numbers, reported in Table 7, clearly show that the bound for λ min (P) is better than that of λ min (A). Details of this calculation have been reported in [7].…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

Gia

Tran

2010

Journal of Complexity

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“…Hence preconditioners are needed to overcome this problem. Recently additive Schwarz preconditioners were used to solve pseudodifferential equations on the unit sphere with rbfs [12,18] and with spherical splines [14,16]. Another kind of preconditioner, the alternate triangular preconditioner, was proposed by Samarskii [17] to solve the Poisson equation with a finite difference method on the unit square.…”

Section: Introductionmentioning

confidence: 99%

Efficient solvers for the shallow water equations on a sphere

Tregubov

Tran

2013

ANZIAMJ

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We present a finite element method using spherical splines to solve the shallow water equations on a sphere involving satellite data. We compare the proposed method with a meshless method using radial basis functions. The use of either radial basis functions or spherical splines leads to ill-conditioned systems of linear equations. To accelerate the solution process we use additive Schwarz and alternate triangular preconditioners. Some numerical experiments are presented to show the effectiveness of both preconditioners.

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“…We developed preconditioners of a similar type for interpolation of scalar functions and pseudo-differential equations on the unit sphere [3,4].…”

Section: Introductionmentioning

confidence: 99%

Additive Schwarz preconditioners for interpolation of divergence-free vector fields on spheres

Gia¹,

Tran²

2011

ANZIAMJ

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The linear system arising from the interpolation problem of surface divergence-free vector fields using radial basis functions tends to be ill-conditioned when the separation radius of the scattered data is small. When the surface under consideration is the unit sphere, we introduce a preconditioner based on the additive Schwarz method to accelerate the solution process. Theoretical estimates for the condition number of the preconditioned matrix are given. Numerical experiments using scattered data from the magsat satellite show the effectiveness of our preconditioner.

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Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Cited by 14 publications

References 29 publications

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

Efficient solvers for the shallow water equations on a sphere

Additive Schwarz preconditioners for interpolation of divergence-free vector fields on spheres

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