Trapezoidal Rule for Computing Supersingular Integral on a Circle

Abstract

The computation of trapezoidal rule for the supersingular integrals on a circle in boundary element methods is discussed. When the singular point coincides with some priori known point, the convergence rate of the trapezoidal rule is higher than the global one which is considered as the superconvergence phenomenon. Then the error functional of density function is derived and the superconvergence phenomenon of composite trapezoidal rule occurs at certain local coordinate of each subinterval. At last, several numerical examples are provided to validate the theoretical analysis and show the efficiency of the algorithms.

Appendix

For the case $p=2$, by the definition of (3), we have

$$\begin{aligned}&\displaystyle \omega _i^{2}(s)\nonumber \\&\quad \displaystyle = \frac{1}{h}\, =\!\!\!\!\!\!\!\int _{x_{i-1}}^{x_{i}} \frac{(x_{i-1}-x)\cos \frac{x-s}{2}}{\sin ^{3}\frac{x-s}{2}}dx \nonumber \\&\qquad \displaystyle +\,\frac{1}{h}\int _{x_{i}}^{x_{i+1}} \frac{(x-x_{i+1})\cos \frac{x-s}{2}}{\sin ^{3}\frac{x-s}{2}}dx \nonumber \\&\quad \displaystyle = \frac{1}{h}\lim _{\epsilon \rightarrow 0} \left\{ \left\{ \int _{x_{i-1}}^{s-\epsilon } +\int _{s+\epsilon }^{x_{i}}\right\} \frac{(x_{i-1}-x)\cos \frac{x-s}{2}}{\sin ^3 \frac{x-s}{2}}\,dx-\frac{-2\epsilon }{\sin ^{2} \frac{\epsilon }{2}}\right\} \nonumber \\&\qquad \displaystyle +\,\frac{1}{h}\int _{x_{i}}^{x_{i+1}} \frac{(x-x_{i+1})\cos \frac{x-s}{2}}{\sin ^{3}\frac{x-s}{2}}dx \nonumber \\&\quad \displaystyle = \frac{1}{h}[\lim _{\epsilon \rightarrow 0} \left\{ -\frac{(x_{i-1}-x)}{\sin ^{2}\frac{x-s}{2}}|_{x_{i-1}}^{s-\epsilon }-\frac{(x_{i-1}-x)}{\sin ^{2}\frac{x-s}{2}}|_{s+\epsilon }^{x_{i}}+\frac{2\epsilon }{\sin ^{2} \frac{\epsilon }{2}}\right\} \nonumber \\&\qquad \displaystyle +\,\, =\!\!\!\!\!\!\!\int _{x_{i-1}}^{x_{i}} \frac{1}{\sin ^{2}\frac{x-s}{2}}dx] \nonumber \\&\qquad \displaystyle +\,\frac{1}{h}[-\frac{(x-x_{i+1})}{\sin ^{2}\frac{x-s}{2}}|_{x_{i}}^{x_{i+1}} \displaystyle +\int _{x_{i}}^{x_{i+1}} \frac{1}{\sin ^{2}\frac{x-s}{2}}dx] \nonumber \\&\quad \displaystyle = \frac{1}{h}[\, =\!\!\!\!\!\!\!\int _{x_{i-1}}^{x_{i}} \frac{1}{\sin ^{2}\frac{x-s}{2}}dx+\int _{x_{i}}^{x_{i+1}} \frac{1}{\sin ^{2}\frac{x-s}{2}}dx] \nonumber \\&\quad \displaystyle = \frac{1}{h}[\lim _{\epsilon \rightarrow 0} \left\{ \left\{ \int _{x_{i-1}}^{s-\epsilon } +\int _{s+\epsilon }^{x_{i}}\right\} \frac{1}{\sin ^2 \frac{x-s}{2}}\,dx-4\cot \frac{\epsilon }{2}\right\} \nonumber \\&\qquad \displaystyle +\int _{x_{i}}^{x_{i+1}} \frac{1}{\sin ^{2}\frac{x-s}{2}}dx] \nonumber \\&\quad \displaystyle = \frac{1}{h}[\lim _{\epsilon \rightarrow 0} \left\{ -2\cot \frac{x-s}{2}|_{x_{i-1}}^{s-\epsilon }-2\cot \frac{x-s}{2}|_{s+\epsilon }^{x_{i}}-4\cot \frac{\epsilon }{2}\right\} \nonumber \\&\qquad \displaystyle -2\cot \frac{x-s}{2}|_{x_{i}}^{x_{i+1}}] \nonumber \\&\quad \displaystyle = \frac{2}{h}\left[ \cot \frac{x_{i+1}-s}{2}+\cot \frac{x_{i-1}-s}{2}-2\cot \frac{x_{i}-s}{2}\right] \nonumber \\&\quad \displaystyle = \frac{2}{h}\left[ \frac{\cos \frac{x_{i+1}-s}{2}\sin \frac{x_{i-1}-s}{2}+\cos \frac{x_{i-1}-s}{2}\sin \frac{x_{i+1}-s}{2}}{\sin \frac{x_{i+1}-s}{2}\sin \frac{x_{i-1}-s}{2}}-2\cot \frac{x_{i}-s}{2}\right] \nonumber \\&\quad \displaystyle = \frac{4}{h}\left[ \frac{\sin (x_{i}-s)}{\cos h-\cos (x_{i}-s)}-\cot \frac{x_{i}-s}{2}\right] , \end{aligned}$$

(80)

where we have used the identity

$$\begin{aligned} x_{i}= & {} x_{i-1}+h,x_{i+1}=x_{i}+h,\nonumber \\ \sin \alpha \sin \beta= & {} -\frac{1}{2}[\cos (\alpha +\beta )-\cos (\alpha -\beta )] \end{aligned}$$

(81)

and

$$\begin{aligned} \sin (\alpha +\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta . \end{aligned}$$

(82)

For the case $p=1$, we have

(83)