Gauss Harmonic Interpolation Formulas for Circular Regions

Johnson, L. Clark; Riess, R. D.

doi:10.1093/imamat/23.2.215

Cited by 3 publications

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“…(ii) For a given ~ in D, use the results in [6] to find the interpolation nodes tl, t2,'", tN on C for the point fl = ~(~). Once the nodes are found, the weights are determined from…”

Section: Numerical Solution For the Formulamentioning

confidence: 99%

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An algorithm for Gauss harmonic formulas

Chen

Johnson

Riess

1984

BIT

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Abstract.Let D be an open, bounded, simply-connected region in R 2 with boundary B. Let (x*,y*) be an arbitrary point of D. This paper constructs an algorithm for computing Gauss harmonic formulas for D and the point (x*,y*). Such formulas approximate a harmonic function at (x*,y*)in terms of a linear combination of its boundary values. Such formulas are useful for approximating the solution of the Dirichlet problem, especially when the problem is to be solved many times at the same point with different boundary values.

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“…(ii) For a given ~ in D, use the results in [6] to find the interpolation nodes tl, t2,'", tN on C for the point fl = ~(~). Once the nodes are found, the weights are determined from…”

Section: Numerical Solution For the Formulamentioning

confidence: 99%

“…Barrow [4,5] discussed similar formulas for the heat equation. Finally Johnson and Riess [6] gave a closed form for formulas on a circular region. The purpose of this paper is to construct a method for computing such formulas for general regions in R 2.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for Gauss harmonic formulas

Chen

Johnson

Riess

1984

BIT

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A Survey of Gauss-Christoffel Quadrature Formulae

Gautschi

1981

E. B. Christoffel

160

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Interpolation formulas for 1-harmonic functions on the unit circle

Djukic,

Mutavdzic,

Mutavdzic-Djukic

2024

Filomat

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Ageneralization of the deeply investigated harmonic functions, known as ?-harmonic functions, have recently gained considerable attention. Similarly to the harmonic functions, an ?-harmonic function u on the unit disc D is uniquely determined by its values on the boundary of the disc ?D. In fact, for any z ? D, the value of u(z) can be given as a contour integral over ?D with a modified Poisson kernel. However, this integral can be difficult to evaluate, or the values on the boundary are known only empirically. In such cases, approximating u(z) with an interpolatory formula, as a weighted sum of values of u at n nodes on ?D, can be an attractive alternative. The nodes and weights are to be chosen so that the degree d of exactness of the formula is maximized. In other words, the formula should be exact for all basis functions for ?-harmonic functions of degree up to d, with d as large as possible. In the case of harmonic functions, it is known that there is an interpolation formula of degree of exactness as large as d = n ? 1. The objective of this paper are formulas of this type for ?-harmonic functions. We will prove that, given n, in this case the degree of exactness cannot be n ? 1, but there is a unique interpolation formula of degree n ? 2. Finally, we will prove convergence of such formulas to u(z) as n ? ?.

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Gauss Harmonic Interpolation Formulas for Circular Regions

Cited by 3 publications

References 0 publications

An algorithm for Gauss harmonic formulas

An algorithm for Gauss harmonic formulas

A Survey of Gauss-Christoffel Quadrature Formulae

Interpolation formulas for 1-harmonic functions on the unit circle

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