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A 1 over r singularity on the reference triangle

Statement

On $T = {(x, y) : 0 \leq y \leq x \leq 1}$ , the weakly singular integral $\int_{T} \frac{d x d y}{\sqrt{x^{2} + y^{2}}} = \log (1 + \sqrt{2}) .$ Under The reference-triangle Duffy map from a vertex, its transformed amplitude is $G (r, s) = (1 + s^{2})^{- 1 / 2}$ with radial weight $r^{0}$ . For every $n \geq 1$ , Weighted product Gauss quadrature after the Duffy transform applies with $α = 1$ , and The Duffy weighted product rule has an explicit bound using pure coordinate derivatives bounds its error by $K_{0} ∥ \partial_{r}^{2 n} G ∥_{\infty} + K_{0} ∥ \partial_{s}^{2 n} G ∥_{\infty}$ with the constants specified there; here $\partial_{r}^{2 n} G = 0$ .

Facts & Assumptions

Given: The reference triangle $T$ and integrand $(x^{2} + y^{2})^{- 1 / 2}$ away from its vertex.

[F1]

The Duffy map turns a vertex singularity into an integrable radial weight: for exponent $α = 1$ and amplitude $q = 1$ , the transformed radial weight is $r^{0}$ and $G (r, s) = (1 + s^{2})^{- 1 / 2}$ .

[F2]

The Duffy weighted product rule has an explicit bound using pure coordinate derivatives: the weighted product rule error is bounded by the two pure derivative norms with its stated constants.

[F3]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : differentiating gives $\frac{d}{d s} \log (s + \sqrt{1 + s^{2}}) = (1 + s^{2})^{- 1 / 2}$ for $s \in [0, 1]$ . The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ evaluates the integral of this continuous derivative by its endpoint increment.

[F4]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and Continuity and derivatives of positive-base real powers: the displayed amplitude is smooth in $s$ because $1 + s^{2}$ is strictly positive.

[F5]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: continuous derivatives on the compact square are bounded.

Proof

1.1F1F3algebra

For $G (r, s) = (1 + s^{2})^{- 1 / 2}$ , the transformed integral in [F1] is $\int_{0}^{1} \int_{0}^{1} G (r, s) d s d r$ . An antiderivative in $s$ is the logarithm in [F3], so the integral equals $\log (1 + \sqrt{2}) - \log 1 = \log (1 + \sqrt{2})$ .

2.1F2F4F5givenalgebra∎

The function $G$ is smooth on the compact square by [F4], and its derivatives are bounded there by [F5]. The Duffy weighted product rule therefore has the error bound [F2]; since $G$ is independent of $r$ , its positive-order $r$ derivative vanishes.

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Sources

Imperial College London, Numerical quadrature, Finite Element course notes, §1.6 (standard reference, not scraped)
Christopher J. Bishop, The Riemann Mapping Theorem, Chapter 3 §3, Gauss-Jacobi quadrature lecture notes (standard reference, not scraped)