Cubature formulae for nearly singular and highly oscillating integrals

Abstract

The paper deals with the approximation of integrals of the type

$$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$

where ${\mathrm {D}}=[-\,1,1]^2$, f is a function defined on ${\mathrm {D}}$ with possible algebraic singularities on $\partial {\mathrm {D}}$, ${\mathbf {w}}$ is the product of two Jacobi weight functions, and the kernel ${\mathbf {K}}$ 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.

Appendix

Now we derive the expression of the cubature rule (2). Recalling the settings

$$\begin{aligned}&\psi _k(z)=\left( \frac{z+1}{2}\right) d-\omega _1+(k-1)d,\quad k=1,2,\dots ,S\\&F_{i,j}({\mathbf {x}})=F\left( \frac{\psi _i(x_1)}{\omega _1},\frac{\psi _j(x_2)}{\omega _1}\right) ,\quad K_{i,j}({\mathbf {x}})={\mathbf {K}}\left( \frac{\psi _i(x_1)}{\omega _1},\frac{\psi _j(x_2)}{\omega _1},\omega \right) \\&w_{i,j}({\mathbf {x}})=w_1\left( \frac{\psi _i(x_1)}{\omega _1}\right) w_2\left( \frac{\psi _j(x_2)}{\omega _1}\right) , \end{aligned}$$

we get

$$\begin{aligned} \mathcal {I}(F,\omega )= & {} \frac{d^2\tau _0}{4}\sum _{i=1}^{S} \sum _{j=1}^{S} \int _{{\mathrm {D}}} F_{i,j}({\mathbf {x}})K_{i,j}({\mathbf {x}}) w_{i,j}({\mathbf {x}}) d{\mathbf {x}}\\= & {} \frac{d^2\tau _0}{4}\left\{ \tau _1\int _{{\mathrm {D}}} F_{1,1}({\mathbf {x}}) K_{1,1}({\mathbf {x}}) U_1({\mathbf {x}})u_2(x_1)u_4(x_2)\right. d{\mathbf {x}}\\&\quad + \tau _2 \int _{{\mathrm {D}}}F_{1,S}({\mathbf {x}})K_{1,S}({\mathbf {x}}) U_2({\mathbf {x}})u_2(x_1) u_3(x_2) d{\mathbf {x}}\\&\quad + \tau _3 \int _{{\mathrm {D}}} F_{S,1}({\mathbf {x}})K_{S,1}({\mathbf {x}}) U_3({\mathbf {x}})u_1(x_1) u_4(x_2) d{\mathbf {x}}\\&\quad + \tau _4 \int _{{\mathrm {D}}} F_{S,S}({\mathbf {x}})K_{S,S}({\mathbf {x}}) U_4({\mathbf {x}}) u_1(x_1) u_3(x_2) d{\mathbf {x}}\\&\quad + \tau _1 \sum _{j=2}^{S-1} \int _{{\mathrm {D}}} F_{1,j}({\mathbf {x}})K_{1,j}({\mathbf {x}}) U_{5,j}({\mathbf {x}}) u_2(x_1)d{\mathbf {x}}\\&\quad + \tau _2 \sum _{i=2}^{S-1} \int _{{\mathrm {D}}} F_{i,S}({\mathbf {x}})K_{i,S}({\mathbf {x}}) U_{6,i}({\mathbf {x}})u_3(x_2)d{\mathbf {x}}\\&\quad + \tau _1 \sum _{i=2}^{S-1} \int _{{\mathrm {D}}} F_{i,1}({\mathbf {x}})K_{i,1}({\mathbf {x}}) U_{7,i}({\mathbf {x}})u_4(x_2)d{\mathbf {x}}\\&\quad + \tau _3 \sum _{j=2}^{S-1} \int _{{\mathrm {D}}} F_{S,j}({\mathbf {x}})K_{S,j}({\mathbf {x}}) U_{8,j}({\mathbf {x}})u_1(x_1)d{\mathbf {x}}\\&+ \left. \tau _1 \sum _{i=2}^{S-1} \sum _{j=2}^{S-1} \int _{{\mathrm {D}}} F_{i,j}({\mathbf {x}})K_{i,j}({\mathbf {x}})U_{9,i,j}({\mathbf {x}}) d{\mathbf {x}}\right\} , \end{aligned}$$

where

$$\begin{aligned} U_1({\mathbf {x}})= & {} v^{\alpha _1,0}\left( \frac{\psi _1(x_1)}{\omega _1}\right) v^{\alpha _2,0}\left( \frac{\psi _1(x_2)}{\omega _1}\right) ,\\ U_2({\mathbf {x}})= & {} v^{\alpha _1,0}\left( \frac{\psi _1(x_1)}{\omega _1}\right) v^{0,\beta _2}\left( \frac{\psi _S(x_2)}{\omega _1}\right) , \\ U_3({\mathbf {x}})= & {} v^{0,\beta _1}\left( \frac{\psi _S(x_1)}{\omega _1}\right) v^{\alpha _2,0}\left( \frac{\psi _1(x_2)}{\omega _1}\right) ,\\ U_4({\mathbf {x}})= & {} v^{0,\beta _1}\left( \frac{\psi _S(x_1)}{\omega _1}\right) v^{0,\beta _2}\left( \frac{\psi _S(x_2)}{\omega _1}\right) ,\\ U_{5,j}({\mathbf {x}})= & {} v^{\alpha _1,0}\left( \frac{\psi _1(x_1)}{\omega _1}\right) w_2\left( \frac{\psi _j(x_2)}{\omega _1}\right) , \\ U_{6,i}({\mathbf {x}})= & {} v^{0,\beta _2}\left( \frac{\psi _S(x_2)}{\omega _1}\right) w_1\left( \frac{\psi _i(x_1)}{\omega _1}\right) ,\\ U_{7,i}({\mathbf {x}})= & {} v^{\alpha _2,0}\left( \frac{\psi _1(x_2)}{\omega _1}\right) w_1\left( \frac{\psi _i(x_1)}{\omega _1}\right) ,\\ U_{8,j}({\mathbf {x}})= & {} v^{0,\beta _1}\left( \frac{\psi _S(x_1)}{\omega _1}\right) w_2\left( \frac{\psi _j(x_2)}{\omega _1}\right) ,\\ U_{9,i,j}({\mathbf {x}})= & {} w_1\left( \frac{\psi _i(x_1)}{\omega _1}\right) w_2\left( \frac{\psi _j(x_2)}{\omega _1}\right) \end{aligned}$$

and

$$\begin{aligned}&\tau _1=\left( \frac{d}{2\omega _1}\right) ^{\beta _1+\beta _2}, \quad \tau _2=\left( \frac{d}{2\omega _1 }\right) ^{\beta _1+\alpha _2}, \quad \tau _3=\left( \frac{d}{2\omega _1 }\right) ^{\alpha _1+\beta _2}, \quad \tau _4=\left( \frac{d}{2\omega _1 }\right) ^{\alpha _1+\alpha _2}.\\&u_0=v^{0,0}, \quad u_1=v^{\alpha _1,0}, \quad u_2=v^{0,\beta _1}, \quad u_3=v^{\alpha _2,0}, \quad u_4=v^{0,\beta _2}. \end{aligned}$$

Then, approximating each integral by the proper Gauss–Jacobi rule depending on the couple of weight functions arising in the integral, according to the notation in (9), we get

$$\begin{aligned} \mathcal {I}(F,\omega )= & {} \frac{d^2\tau _0}{4}\left\{ \tau _1 \mathcal {G}_{m,m}^{(u_2,u_4)}\left( F_{1,1}K_{1,1} U_1\right) + \tau _2 \mathcal {G}_{m,m}^{(u_2,u_3)}\left( F_{1,S}K_{1,S}U_2\right) \right. \nonumber \\&\quad + \tau _3 \mathcal {G}_{m,m}^{(u_1,u_4)}\left( F_{S,1}K_{S,1} U_3\right) + \tau _4 \mathcal {G}_{m,m}^{(u_1,u_3)}\left( F_{S,S}K_{S,S} U_4\right) \nonumber \\&\quad + \tau _1 \sum _{j=2}^{S-1} \mathcal {G}_{m,m}^{(u_2,u_0)}\left( F_{1,j}K_{1,j} U_{5,j}\right) + \tau _2 \sum _{i=2}^{S-1} \mathcal {G}_{m,m}^{(u_0,u_3)}\left( F_{i,S}K_{i,S} U_{6,i} \right) \nonumber \\&\quad + \tau _1 \sum _{i=2}^{S-1} \mathcal {G}_{m,m}^{(u_0,u_4)}\left( F_{i,1}K_{i,1} U_{7,i}\right) + \tau _3 \sum _{j=2}^{S-1} \mathcal {G}_{m,m}^{(u_1,u_0)}\left( F_{S,j}K_{S,j} U_{8,j}\right) \nonumber \\&\quad + \left. \tau _1 \sum _{i=2}^{S-1} \sum _{j=2}^{S-1} \mathcal {G}_{m,m}^{(u_0,u_0)}\left( F_{i,j}K_{i,j} U_{9,i,j}\right) \right\} =:\varPsi _{m,m}(F,\omega )+\mathcal {R}^\varPsi _{m,m}(F,\omega ).\nonumber \\ \end{aligned}$$

(38)