# Cubature formulae for nearly singular and highly oscillating integrals

Cubature formulae for nearly singular and highly oscillating integrals The paper deals with the approximation of integrals of the type \begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned} I ( f ; t ) = ∫ D f ( x ) K ( x , t ) w ( x ) d x , x = ( x 1 , x 2 ) , t ∈ T ⊆ R p , p ∈ { 1 , 2 } where $${\mathrm {D}}=[-\,1,1]^2$$ D = [ - 1 , 1 ] 2 , f is a function defined on $${\mathrm {D}}$$ D with possible algebraic singularities on $$\partial {\mathrm {D}}$$ ∂ D , $${\mathbf {w}}$$ w is the product of two Jacobi weight functions, and the kernel $${\mathbf {K}}$$ K can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calcolo Springer Journals

# Cubature formulae for nearly singular and highly oscillating integrals

, Volume 55 (1) – Feb 7, 2018
33 pages

/lp/springer_journal/cubature-formulae-for-nearly-singular-and-highly-oscillating-integrals-k30JD0Gn9C
Publisher
Springer Journals
Subject
Mathematics; Numerical Analysis; Theory of Computation
ISSN
0008-0624
eISSN
1126-5434
D.O.I.
10.1007/s10092-018-0243-x
Publisher site
See Article on Publisher Site

### Abstract

The paper deals with the approximation of integrals of the type \begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned} I ( f ; t ) = ∫ D f ( x ) K ( x , t ) w ( x ) d x , x = ( x 1 , x 2 ) , t ∈ T ⊆ R p , p ∈ { 1 , 2 } where $${\mathrm {D}}=[-\,1,1]^2$$ D = [ - 1 , 1 ] 2 , f is a function defined on $${\mathrm {D}}$$ D with possible algebraic singularities on $$\partial {\mathrm {D}}$$ ∂ D , $${\mathbf {w}}$$ w is the product of two Jacobi weight functions, and the kernel $${\mathbf {K}}$$ K can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed.

### Journal

CalcoloSpringer Journals

Published: Feb 7, 2018

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