A Fast Fourier–Galerkin Method for Solving Singular Boundary Integral Equations

Cai, Haotao; Xu, Yuesheng

doi:10.1137/070703478

Cited by 27 publications

(38 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For the dense matrix B, since the kernel function b is smooth, we use the hyperbolic cross approximate method in [6,14,15,19] to compress matrix B into a sparse matrixB without loss of critical information encoded in the matrix. Specifically, we introduce an index set…”

Section: The Matrix Truncation Strategy and Its Analysismentioning

confidence: 99%

“…If we use the symbol M (G) to denote the number of the nonzero entries for any matrix G, then Theorem 3.1 in [6] shows that M (B) is O(n log 2 n).…”

Section: The Matrix Truncation Strategy and Its Analysismentioning

confidence: 99%

“…This fast truncated Fourier-Galerkin method preserves the optimal order of the approximate solution, and leads to a semi-discrete linear system with a sparse coefficient matrix. In [14], they continue the theme of [6] and obtain a fast fully discrete Fourier-Galerkin method. This method generates a fully discrete sparse linear system with a nearly linear computational complexity, and preserves the optimal order of the approximate solution.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome this shortcoming, in [6], a fast truncated Fourier-Galerkin method is presented for solving the singular boundary integral equations, which are induced from the two-dimensional interior Dirichlet and Neumann boundary value problem for Laplace equation with a smooth boundary. This fast truncated Fourier-Galerkin method preserves the optimal order of the approximate solution, and leads to a semi-discrete linear system with a sparse coefficient matrix.…”

Section: Introductionmentioning

confidence: 99%

“…This method generates a fully discrete sparse linear system with a nearly linear computational complexity, and preserves the optimal order of the approximate solution. In [24], following the ideas coming from [6,14], a fast fully discrete Fourier-Galerkin method is used for solving first-kind logarithmic-kernel integral equation, induced from the two-dimensional Dirichlet problem for the Laplace equation on open arcs. In this paper, we shall continue the work of [6,14,24] and develop a fast numerical solution for the first kind boundary integral equation, arising from the interior Dirichlet problem for Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

Cai

2012

Bit Numer Math

Self Cite

View full text Add to dashboard Cite

The main purpose of this paper is to develop a fast numerical method for solving the first kind boundary integral equation, arising from the two-dimensional interior Dirichlet boundary value problem for the Helmholtz equation with a smooth boundary. This method leads to a fully discrete linear system with a sparse coefficient matrix. We observe that it requires a nearly linear computational cost to produce and then solve this system, and the corresponding approximate solution obtained by this proposed method preserves the optimal convergence order. One numerical example demonstrates the efficiency and accuracy of the proposed method.

show abstract

Section: The Matrix Truncation Strategy and Its Analysismentioning

confidence: 99%

“…If we use the symbol M (G) to denote the number of the nonzero entries for any matrix G, then Theorem 3.1 in [6] shows that M (B) is O(n log 2 n).…”

Section: The Matrix Truncation Strategy and Its Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

Cai

2012

Bit Numer Math

Self Cite

View full text Add to dashboard Cite

show abstract

Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration

Vondřejc

2016

Int. J. Numer. Meth. Engng

View full text Add to dashboard Cite

Moulinec and Suquet introduced FFT-based homogenization in 1994, and 20 years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author's (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec-Suquet algorithm, was used to evaluate guaranteed upper-lower bounds on homogenized coefficients incorporating a closed-form doublegrid quadrature. Here, these concepts, based on the primal and dual formulations, are employed for the Ga scheme. For the same computational effort, the Ga outperforms the GaNi with more accurate guaranteed bounds and more predictable numerical behaviors. The quadrature technique leading to block-sparse linear systems is extended here to materials defined via high-resolution images in a way that allows for effective treatment using the FFT. Memory demands are reduced by a reformulation of the double-grid scheme to the original grid scheme using FFT shifts. Minimization of the bounds during iterations of conjugate gradients is effective, particularly when incorporating a solution from a coarser grid. The methodology presented here for the scalar linear elliptic problem could be extended to more complex frameworks.

show abstract

Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations

Xiang-ling

Wang

2013

J Sci Comput

View full text Add to dashboard Cite

A Fast Fourier–Galerkin Method for Solving Singular Boundary Integral Equations

Cited by 27 publications

References 18 publications

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration

Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations

Contact Info

Product

Resources

About