Source nodes on elliptic pseudo-boundaries in the method of fundamental solutions for Laplace’s equation; selection of pseudo-boundaries

Dou, Fangfang; Zhang, Li Ping; Li, Zi Cai; Chen, C. S.

doi:10.1016/j.cam.2020.112861

Cited by 17 publications

(8 citation statements)

References 8 publications

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“…In Figures 1 and 2, the curves of Lower and Upper in (146) and (147) are approximately straight‐lines. This confirms that the Cond grows exponentially; the same as the standard MFS in References 5, 7, and 11. Since estimates of Cond are derived via the Vandermonde‐wise matrix

\hat{\mathbf{E}}

(or

\mathbf{E}

), numerical Cond are closer to

\mathrm{Cond}\left(\mathbf{E}\right)

(or

\mathrm{Cond}\left(\hat{\mathbf{E}}\right)

) in Figures 1 and 2.…”

Section: Numerical Experimentssupporting

confidence: 77%

“…Compared upper bounds to lower bounds, there occurs an additional exponential factor

O\left({\sqrt{2}}^N\right)

. Both lower and upper bounds of Cond provide a complete knowledge of stability of the new algorithms in this article and Reference 14; they are intriguing in stability analysis for the MFS in References 5, 7, and 11. All numerical Cond agree with the theoretical estimates. We derive lower and upper bounds of Cond for the standard Vandermonde matrix by the uniform

{x}_i\in \left[a,b\right]

with

0<a<b<1

.…”

Section: Discussionsupporting

confidence: 58%

“…Comparing (13) with (12), there occurs an additional exponential rate

O\left({\sqrt{2}}^N\right)

of Cond for

N\gg 1

. The exponential growth of Cond is confirmed from (12) and (13), which is the same as that in the standard MFS but lower than the super‐exponential growth of the boundary knot method (BKM) in Zhang et al 8 Both lower and upper bound estimates of Cond in this article are intriguing in the stability study for the MFS in References 5, 7, 9, and 11. All numerical condition numbers are coincident with the lower and upper bound estimates.…”

Section: Introductionsupporting

confidence: 64%

“…In the standard MFS, the source nodes are located on a closed contour outside the domain boundary

\Gamma \left(=\partial S\right)

, called pseudo‐boundary. Error and stability analysis is made for circular/elliptic pseudo‐boundaries in References 4‐11. We study new locations of source nodes along pseudo radial‐lines outside

\Gamma

.…”

Section: Introductionmentioning

confidence: 99%

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Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

Zhang

Lee

et al. 2022

Numerical Linear Algebra App

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Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain S. In the standard MFS, the source nodes are located on a closed contour outside the domain boundary Γ(= 𝜕S), which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside Γ. Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes x i ∈ [a, b] with 0 < a < b < 1. The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as {ln |PQ i |}) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes x i ∈ [a, b] with 0 < a < b < 1 may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.

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\hat{\mathbf{E}}

(or

\mathbf{E}

), numerical Cond are closer to

\mathrm{Cond}\left(\mathbf{E}\right)

(or

\mathrm{Cond}\left(\hat{\mathbf{E}}\right)

) in Figures 1 and 2.…”

Section: Numerical Experimentssupporting

confidence: 77%

“…Compared upper bounds to lower bounds, there occurs an additional exponential factor

O\left({\sqrt{2}}^N\right)

{x}_i\in \left[a,b\right]

with

0<a<b<1

.…”

Section: Discussionsupporting

confidence: 58%

“…Comparing (13) with (12), there occurs an additional exponential rate

O\left({\sqrt{2}}^N\right)

of Cond for

N\gg 1

Section: Introductionsupporting

confidence: 64%

“…In the standard MFS, the source nodes are located on a closed contour outside the domain boundary

\Gamma \left(=\partial S\right)

\Gamma

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

Zhang

Lee

et al. 2022

Numerical Linear Algebra App

Self Cite

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show abstract

“…source points and the ill conditioning. The location of the source points has been widely addressed in the literature ( [1,2,3,13,19,23,24,26,27,33]) and several choices have been advocated to be effective. Some works have proposed techniques to alleviate the ill conditioning if the MFS [9,10,17,37], but none of these approaches seem to completely solve the problem of ill conditioning.…”

Section: Introductionmentioning

confidence: 99%

A well conditioned Method of Fundamental Solutions

Antunes¹

2021

Preprint

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The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are illconditioned and this may prevent to achieve high accuracy.In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.

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Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

Zhang,

Li,

Lee

et al. 2024

Numerical Methods Partial

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Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.

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Source nodes on elliptic pseudo-boundaries in the method of fundamental solutions for Laplace’s equation; selection of pseudo-boundaries

Cited by 17 publications

References 8 publications

Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

A well conditioned Method of Fundamental Solutions

Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

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