Multi-dimensional quadrature of singular and discontinuous functions.

*(English)*Zbl 1021.65010This 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.

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.

Reviewer: Jesus Illán González (Vigo)

##### 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) |