Spectral multidomain technique with Local Fourier Basis II: Decomposition into cells

Vozovoi, L.; Israeli, M.; Averbuch, Amir

doi:10.1007/bf01575035

Cited by 17 publications

(11 citation statements)

References 3 publications

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“…(1.1) is extended to a larger domain and it is replaced by a new function which coincides with f in the original domain and it is periodic together with a certain number of its derivatives in the larger domain [4,18]. This procedure is based on the local Fourier basis method [14,19] and it uses the folding operation as it is described in [1,13].…”

Section: Introductionmentioning

confidence: 99%

A Fast 3D Poisson Solver of Arbitrary Order Accuracy

Braverman

Israeli

Averbuch

et al. 1998

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

A Fast 3D Poisson Solver of Arbitrary Order Accuracy

Braverman

Israeli

Averbuch

et al. 1998

Journal of Computational Physics

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“…In this case the implementation of Algorithm IV involves the interpolation step (from Chebyshev to Fourier nodes and vice versa) in order to combine u p and u h . An alternative approach for computation of the particular solution implemented in this paper, consists of expanding the source function into a finite trigonometric series along with a smoothing procedure near the boundaries (the LFB method of [5], [14]). The computational complexity of this method scales like…”

Section: The Analytic Case the Function Umentioning

confidence: 99%

“…Appendix B. The detailed analysis of the LFB technique, as applied for the solution of problems in multidomain regions, is given in [5], [14]. Here we describe briefly only a part of this technique required for the solution of nonperiodic ODEs in one domain.…”

Section: Fig 3 the Complete Solution Of (59)mentioning

confidence: 99%

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A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

Averbuch¹,

Israeli²,

Vozovoi³

1998

SIAM J. Sci. Comput.

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Abstract. In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems.The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms.The solution of a boundary value problem is obtained in a series form in O(N log N ) floating point operations, where N 2 is the number of grid nodes. Evaluating the solution at all N 2 interior points requires O(N 2 log N ) operations.Key words. boundary value problem, Poisson equation, rectangular region, spectral method, corner discontinuities, polynomial subtraction AMS subject classifications. 35J05, 45L10, 65P05PII. S1064827595288589 1.Introduction. An important step in the development of fast numerical solvers for elliptic equations in complicated domains is an algorithm for the solution of boundary value problems for constant coefficient elliptic equations in rectangular regions. After solving a nonhomogeneous equation with some "convenient" boundary conditions, one must solve, in a correction step, the homogeneous problem with specified (Dirichlet or Neumann) boundary conditions. As a resulting boundary distribution may not satisfy the required compatibility conditions at the corner points, singularities may arise in the process of the solution, thus destroying the convergence rate even if the final solution is smooth.When the computational region is discretized on a N × N grid using some loworder (finite difference or finite element) scheme, the resulting system of linear algebraic equations is represented by a sparse matrix. Such a matrix can be inverted in O(N 2 ) (or O(N 2 log 2 N ) arithmetic operations using a "fast solver" [4]. However, since the method is of low order, the resolution N must be very large if a high accuracy is desired.Application of high-order (pseudo) spectral methods, based on global expansions into orthogonal polynomials, e.g., Chebyshev polynomials, to the solution of elliptic equations, results in a full matrix problem. The cost of inverting such a matrix using the best current algorithms is O(N 3 ) operations [2]. Besides, the accuracy decreases considerably as the dimension N 2 of the system grows due to accumulation of roundoff errors.When using the Chebyshev method, the fast computation of the expansion coefficients requires that the problem be discretized on a nonuniform grid. For timedependent problems, when elliptic equations arise due to a time-discretization pro-

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“…(1.1) or (1.2) is extended to a larger domain and replaced by a new function which coincides with f in the original domain but the periodic extension of the larger domain has a certain number of continuous derivatives [17,4]. The extension procedure is based on the Local Fourier Basis method [12,18] which employs folding functions as described and tested in [6].…”

Section: Introductionmentioning

confidence: 99%

A Hierarchical 3-D Direct Helmholtz Solver by Domain Decomposition and Modified Fourier Method

Braverman

Israeli²,

Averbuch³

2005

SIAM J. Sci. Comput.

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The paper contains a non-iterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput. 20 (1999Comput. 20 ( ), 2237Comput. 20 ( -2260. This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N 3 log N ) operations, where N is the number of grid points in each direction.

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Spectral multidomain technique with Local Fourier Basis II: Decomposition into cells

Cited by 17 publications

References 3 publications

A Fast 3D Poisson Solver of Arbitrary Order Accuracy

A Fast 3D Poisson Solver of Arbitrary Order Accuracy

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

A Hierarchical 3-D Direct Helmholtz Solver by Domain Decomposition and Modified Fourier Method

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