A frequency-domain parallel method for the numerical approximation of parabolic problems

Lee, Chang-Ock; Lee, Jongwoo; Sheen, Dongwoo; Yeom, Yongjin

doi:10.1016/s0045-7825(98)00168-6

Cited by 11 publications

(4 citation statements)

References 4 publications

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“…where is some fixed parameter in frequency domain. It is not difficult to check the existence and uniqueness of the problem (4) with an application of the Lax-Milgram lemma; see [17]. Proposition 1.…”

Section: Fourier Transform and Multiscale Asymptotic Expansionsmentioning

confidence: 99%

Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

Wang

Duan

Gao

2014

Mathematical Problems in Engineering

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An efficient parallel multiscale numerical algorithm is proposed for a parabolic equation with rapidly oscillating coefficients representing heat conduction in composite material with periodic configuration. Instead of following the classical multiscale asymptotic expansion method, the Fourier transform in time is first applied to obtain a set of complex-valued elliptic problems in frequency domain. The multiscale asymptotic analysis is presented and multiscale asymptotic solutions are obtained in frequency domain which can be solved in parallel essentially without data communications. The inverse Fourier transform will then recover the approximate solution in time domain. Convergence result is established. Finally, a novel parallel multiscale FEM algorithm is proposed and some numerical examples are reported.

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Section: Fourier Transform and Multiscale Asymptotic Expansionsmentioning

confidence: 99%

Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

Wang

Duan

Gao

2014

Mathematical Problems in Engineering

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“…Then, the Fourier transform p satisfies the following set of elliptic problems depending on ω: for all ω ∈ R The approximate solution for the problem (1.1) was obtained by time stepping methods such as backward Euler and Crank-Nicolson methods traditionally. In recent a natural parallel algorithm which does not require any significant communication costs was introduced by transforming the parabolic problem (1.1) in the space-time domain into the independent elliptic problems (1.2) in the space-frequency domain [11,12,15,20,21]. See [22,23,27,28] for the Laplace transformation.…”

Section: Introductionmentioning

confidence: 99%

Least-Squares Spectral Collocation Parallel Methods for Parabolic Problems

Seo¹,

Shin²

2015

Honam Mathematical Journal

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Abstract. In this paper, we study the first-order system leastsquares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

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“…) has been considered in [4,3,19,11,13,16,15,14,12], while Fourier-Laplace transformation has been applied in [20].…”

Section: S ) !mentioning

confidence: 99%

“…After these works, with the aid of several coworkers the author has been developing the theory and extending its applications to viscoelasticity, parabolic problems, and linearized Navier-Stokes equations. See [5,11,16,20,15,14,12,20]. This paper essentially surveys in brief such approaches for solving time-dependent linear initial-boundary value problems via Fourier-Laplace transformation.…”

Section: Introductionmentioning

confidence: 99%

Parallel Methods for Solving Time-Dependent Problems Using the Fourier-Laplace Transformation

Sheen

Lecture Notes in Physics

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In this paper we summarize recent progresses on the parallel method for solving time-dependent problems using the Fourier-Laplace transformation. Instead of solving the time-dependent problems in the space-time domain, we solve them as follows. First take the Fourier-Laplace transformation of given problems originally set in the space-time domain, and consider the corresponding problems in the space-frequency domain which form a set of indefinite, complex-valued elliptic problems. Such problems are solved in a natural parallel manner since each problem is independent of others. The Fourier-Laplace inversion formula will then recover the solution in the space-time domain.

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A frequency-domain parallel method for the numerical approximation of parabolic problems

Cited by 11 publications

References 4 publications

Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

Least-Squares Spectral Collocation Parallel Methods for Parabolic Problems

Parallel Methods for Solving Time-Dependent Problems Using the Fourier-Laplace Transformation

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