Numerical quadrature for singular integrals on fractals
We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integratio...
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| Vydané v: | Numerical algorithms Ročník 92; číslo 4; s. 2071 - 2124 |
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| Jazyk: | English |
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01.04.2023
Springer Nature B.V |
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| Abstract | We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain
Γ
⊂
R
n
is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on
Γ
, including in particular the Hausdorff measure
H
d
restricted to
Γ
, where
d
is the Hausdorff dimension of
Γ
. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over
Γ
are decomposed into sums of integrals over suitable partitions of
Γ
into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. |
|---|---|
| AbstractList | We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain
Γ
⊂
R
n
is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on
Γ
, including in particular the Hausdorff measure
H
d
restricted to
Γ
, where
d
is the Hausdorff dimension of
Γ
. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over
Γ
are decomposed into sums of integrals over suitable partitions of
Γ
into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $$\Gamma \subset \mathbb {R}^n$$ Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on $$\Gamma$$ Γ , including in particular the Hausdorff measure $$\mathcal {H}^d$$ H d restricted to $$\Gamma$$ Γ , where d is the Hausdorff dimension of $$\Gamma$$ Γ . Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over $$\Gamma$$ Γ are decomposed into sums of integrals over suitable partitions of $$\Gamma$$ Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ⊂Rn is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Γ, including in particular the Hausdorff measure Hd restricted to Γ, where d is the Hausdorff dimension of Γ. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Γ are decomposed into sums of integrals over suitable partitions of Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. |
| Author | Moiola, Andrea Gibbs, Andrew Hewett, David |
| Author_xml | – sequence: 1 givenname: Andrew orcidid: 0000-0002-2934-008X surname: Gibbs fullname: Gibbs, Andrew email: andrew.gibbs@ucl.ac.uk organization: Department of Mathematics, University College London – sequence: 2 givenname: David surname: Hewett fullname: Hewett, David organization: Department of Mathematics, University College London – sequence: 3 givenname: Andrea surname: Moiola fullname: Moiola, Andrea organization: Dipartimento di Matematica, Università degli studi di Pavia |
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| Cites_doi | 10.1007/BF02437506 10.1007/s13373-013-0041-3 10.1007/s00041-011-9202-5 10.4236/wet.2011.22015 10.1090/S0002-9947-98-02218-1 10.1137/17M1131933 10.1007/s00211-021-01182-y 10.1109/8.664115 10.1512/iumj.1981.30.30055 10.1103/PhysRevA.36.920 10.1007/BF02476031 10.1090/S0025-5718-1982-0669644-3 10.1017/S0334270000002757 10.1016/j.jmaa.2004.10.059 10.1090/S0002-9947-99-01982-0 10.1137/S0036141096306911 10.1007/s11075-007-9111-5 10.1016/j.jmaa.2019.01.036 10.1109/8.475113 10.1063/1.1664597 10.1007/BF02878985 10.1007/978-3-0348-0034-1 10.1080/02786820117868 10.1007/s00023-006-0308-2 10.1142/S0218348X09004296 10.1017/CBO9780511623813 10.1007/978-1-4899-6886-9_5 10.1007/s00041-001-4001-z 10.1007/BF01889599 |
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Γ
⊂
R
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is... We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $$\Gamma... We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ⊂Rn is... |
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| SubjectTerms | Acoustic waves Algebra Algorithms Approximation Computer Science Diameters Fractal analysis Fractals Galerkin method Helmholtz equations Integral equations Integrals Numeric Computing Numerical Analysis Original Paper Quadratures Self-similarity Subtraction Theory of Computation Wave scattering |
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