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Multi-dimensional quadrature of singular and discontinuous functions. (English) Zbl 1021.65010
This paper is concerned with the problem of evaluating integrals with non-smooth integrands $$F:\Omega\subset\mathbb R^2\to\mathbb R$$. The singularities of $$F$$ are all located in a given curve $$\gamma\in \Omega$$ which separates $$\Omega$$ into two regions $$\Omega_1$$ and $$\Omega_2$$.
The class of integrands is described as follows. Let $$d$$ be the signed distance function in $$\Omega$$ which is defined as $$d(\mathbf{x})=(-1)^{i+1}$$dist$$(\mathbf{x},\gamma)$$ if $$\mathbf{x}\in \Omega_i$$, $$i=1,2$$. Let $$G(\mathbf{x})$$ be an arbitrary smooth function in $$\Omega$$. The functions $$F(\mathbf{x})$$ to be integrated have the form $$F(\mathbf{x})=f(d(\mathbf{x}))G(\mathbf{x})$$, where $$f$$ is one of the two following. 1. $$f(t)=\delta(t)$$, where $$\delta$$ is the Dirac delta function (singular functions). 2. $$f(t)=H(t)$$, where $$H(t)$$ is the Heaviside function (discontinuous functions).
In general, the author suggests an approach which consists in a regularization of the integrand by introducing a smooth approximating function $$f_w(d(\mathbf{x}))$$ to the function $$f(d(\mathbf{x}))$$. Thus, the whole error in the integration of $$F(\mathbf{x})$$ is the sum $$E_{G}(f)=E_{w,G}(f_w)+E_{quad,G}(f_w)$$, where $$E_{w,G}(f_w)$$ is the analytical error made by replacing $$f(d(\mathbf{x}))$$ with $$f_w(d(\mathbf{x}))$$, and $$E_{quad,G}(f_w)$$ is the numerical error made by applying a quadrature procedure to the integral of $$f_w(d(\mathbf{x}))$$.
By introducing the moments of a function, some results on the analytical error are obtained for both types of integrands. Moreover, regularization techniques to problems for which $$\gamma$$ is close to the boundary, are discussed.
Numerical tests are shown to illustrate the behavior of the method.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 41A55 Approximate quadratures 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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