Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries

Zhang, Liping; Li, Zi‐Cai; Lee, Ming‐Gong; Huang, Hung‐Tsai

doi:10.1007/s11075-021-01150-5

Cited by 5 publications

(7 citation statements)

References 20 publications

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“…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: 62%

“…where N = 2n + 2, M ≥ N, and Φ − i (𝜌, 𝜃) and Φ + i (𝜌, 𝜃) are also given in (10) and (11). Equation ( 77) is denoted by an overdetermined system,…”

Section: Lemma 6 Choose the Uniform Nodes Xmentioning

confidence: 99%

“…where the notation, "A ≍ B", denotes that there exist two constants c 0 (> 0) and C independent of n such that 0 < c 0 B ≤ A ≤ CB. By following the techniques, 11 we have the following lemma.…”

Section: Lemma 6 Choose the Uniform Nodes Xmentioning

confidence: 99%

“…Moreover, when a quadrature rule is chosen (e.g., the trapezoidal rule), there also exist the equivalences,

{\hat{\int}}_a^b{g}_n^2(x)\kern0.3em dx\asymp {\int}_a^b{g}_n^2(x)\kern0.3em dx,\kern1em {\hat{\int}}_a^b{\left({g}_n^{\ast }(x)\right)}^2\kern0.3em dx\asymp {\int}_a^b{\left({g}_n^{\ast }(x)\right)}^2\kern0.3em dx,\kern0.5em

where the notation, “

A\asymp B

”, denotes that there exist two constants

{c}_0\left(>0\right)

and

C

independent of

n

such that

0<{c}_0B\le A\le CB

. By following the techniques, 11 we have the following lemma.…”

Section: Upper Bound Estimates Of Cond For Variant Case By Case IImentioning

confidence: 99%

“…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

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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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“…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: 62%

“…where N = 2n + 2, M ≥ N, and Φ − i (𝜌, 𝜃) and Φ + i (𝜌, 𝜃) are also given in (10) and (11). Equation ( 77) is denoted by an overdetermined system,…”

Section: Lemma 6 Choose the Uniform Nodes Xmentioning

confidence: 99%

Section: Lemma 6 Choose the Uniform Nodes Xmentioning

confidence: 99%

“…Moreover, when a quadrature rule is chosen (e.g., the trapezoidal rule), there also exist the equivalences,

{\hat{\int}}_a^b{g}_n^2(x)\kern0.3em dx\asymp {\int}_a^b{g}_n^2(x)\kern0.3em dx,\kern1em {\hat{\int}}_a^b{\left({g}_n^{\ast }(x)\right)}^2\kern0.3em dx\asymp {\int}_a^b{\left({g}_n^{\ast }(x)\right)}^2\kern0.3em dx,\kern0.5em

where the notation, “

A\asymp B

”, denotes that there exist two constants

{c}_0\left(>0\right)

and

C

independent of

n

such that

0<{c}_0B\le A\le CB

. By following the techniques, 11 we have the following lemma.…”

Section: Upper Bound Estimates Of Cond For Variant Case By Case IImentioning

confidence: 99%

“…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 3 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

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Spurious eigenvalue-free algorithms of the method of fundamental solutions for solving the Helmholtz equation in bounded multiply connected domains

Zhang

Wei

et al. 2022

Numer Algor

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A data structure for finite bounded acyclic categories has been built, which is useful to encode and manipulate abstract orientable incidence structure. It can be represented as a directed acyclic multigraph with weighted edges, where the weighs encode the algebraic structure between edges. The fundamental operations on this data structure are investigated from geometrical, categorical and programming perspectives.

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New locations of source nodes for method of fundamental solutions solving Laplace’s equation; pseudo radial-lines

Zhang

Huang

et al. 2022

Engineering Analysis with Boundary Elements

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Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries

Cited by 5 publications

References 20 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

Spurious eigenvalue-free algorithms of the method of fundamental solutions for solving the Helmholtz equation in bounded multiply connected domains

New locations of source nodes for method of fundamental solutions solving Laplace’s equation; pseudo radial-lines

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