A Modified Trapezoidal Rule for a Class of Weakly Singular Integrals in n Dimensions

Abstract

In this paper we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\mathbb{R}^n}\phi(x)s(x)dx$ in $n$ dimensions, where $\phi\in C_c^N(\mathbb{R}^n)$ for some sufficiently large $N$ and $s$ is the weakly singular kernel. The admissible class of weakly singular kernel requires $s$ satisfies dilation and symmetry properties and is large enough to contain functions of the form $\frac{P(x)}{|x|^r}$ where $r > 0$ and $P(x)$ is any monomials such that $\text{deg} P < r < \text{deg} P + n$. The modified trapezoidal rule is the singularity-punctured trapezoidal rule added by correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule exactly evaluates some monomials and solving corresponding linear systems. A long-standing difficulty of these type of methods is establishing the non-singularity of the linear system, despite strong numerical evidences. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence.