Spectral methods for hyperbolic initial boundary value problems on parallel computers

Dutt, Pravir; Bedekar, Smita

doi:10.1016/s0377-0427(00)00535-5

Cited by 15 publications

(7 citation statements)

References 9 publications

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“…Hence, the proposed method is cheap and efficient to compute the residual in comparison of h − p finite element method. For more details, we refer to [13,32]. Specifically, the comparison between h − p fem and LSSEM for 2 − D elliptic problems is discussed in Section 5 in [32].…”

Section: Symmetric Formulationmentioning

confidence: 99%

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Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Khan

2020

J Sci Comput

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In this paper, a least-squares spectral method and a non-conforming least-squares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a suitable preconditioner and show that the condition number of the preconditioned system is O((ln W) 2) (where W is degree of polynomials). Then we establish the error estimates of the proposed method. Specific numerical results based on linear elasticity problem, fourth order problem and sixth order problem are discussed to reflect the efficiency of the proposed method. Keywords Least-squares methods • Non-conforming methods • Spectral element method • Linear elliptic system in three dimensional space • Fourth order problems • Sixth order problem • Linear elasticity problem • Preconditioner • Exponential accuracy Dedicated to Professor Chandra Shekhar Upadhyay on the occasion of his 50th birthday Research is supported by EPSRC Grant EP/P013317/1.

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Section: Symmetric Formulationmentioning

confidence: 99%

“…The process is similar to those for elliptic equations (for more details, see also [14,32]). Thus, we give a brief overview to compute numerical scheme and refer to [13,14,32] for more details.…”

Section: Appendixmentioning

confidence: 99%

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Khan

2020

J Sci Comput

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“…Moreover we need to compute two global scalars to update the approximate solution and the search direction. Hence inter-processor communication is quite small [15,16,19] (see Fig. 2).…”

Section: Parallelization and Preconditioningmentioning

confidence: 99%

Nonconforming Least-Squares Spectral Element Method for European Options

Khan

Dutt

Upadhyay

2015

Computers & Mathematics with Applications

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a b s t r a c tSeveral methods have been proposed in the literature for solving the Black-Scholes equation for European Options. The method proposed in the current study achieves spectral accuracy in both space and time. The method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation and initial condition in different Sobolev norms, and a term which measures the jump in the function and its derivatives across inter-element boundaries in appropriate fractional Sobolev norms. To obtain values of the solution and its derivatives the initial condition is mollified and the computed solution is post processed. Error estimates are obtained for this method. Specific numerical examples are given to show the efficiency of this method.

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“…, v p N,j ). In [4] it has been shown that the residual in the normal equations can be computed in O(p 3 ) operations without having to compute mass and stiffness matrices.…”

Section: Parallelization and Pre-conditioningmentioning

confidence: 99%

“…In [4] a spectral method to solve hyperbolic initial boundary value problems on parallel computers in one space dimension has been proposed. In [11] these results are generalized to a domain in several dimensions which is divided into subdomains.…”

Section: Introductionmentioning

confidence: 99%

Domain decomposition methods for hyperbolic problems

Dutt

Lamba

2009

Proc Math Sci

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In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the L 2 norms of the residuals and a term which is the sum of the squares of the L 2 norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method.The Schwarz waveform relaxation method is often employed to solve hyperbolic problems using domain decomposition but this technique faces difficulties if the system becomes characteristic at the inter-element boundaries. By making the inter-element boundaries move faster than the fastest wave speed associated with the hyperbolic system we are able to overcome this problem.

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Spectral methods for hyperbolic initial boundary value problems on parallel computers

Cited by 15 publications

References 9 publications

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Nonconforming Least-Squares Spectral Element Method for European Options

Domain decomposition methods for hyperbolic problems

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