Verification of singularity order of BEM kernels

According to (Erichsen and Sauter 1998), kernel functions in BEM are Gâteaux derivatives of the fundamental solution of the underlying boundary value problem. For elliptic boundary value problems, a kernel function \(k(x,y)\) has the general form

\[k(x,y)=\sum_{\abs{\alpha}=t}^{t+a} S_{\alpha}(x,y)\eta_{\alpha}(\norm{x-y})\frac{(y-x)^{\alpha}}{\norm{y-x}^{\sigma+t}},\]

where \(S_{\alpha}: \Gamma\times\Gamma \rightarrow \mathbb{C}\) are analytic on each pair of smooth surface patches of \(\Gamma\). The scalar function \(\eta_{\alpha}\) is also analytic. \(\alpha\in \mathbb{N}_0^3\) is a multi-index and \(\sigma\) is defined as the singularity order of \(k(x,y)\).

When \(\sigma\leq 2\), the kernel has Cauchy singularity. This means the integral formed with this kernel exists in the sense of Cauchy principal value, which can be evaluated by first removing an open ball \(B_{\varepsilon}(x_0)\) around the singular point \(x_0\) from the integration domain and then taking the limit \(\varepsilon \rightarrow 0\). When \(\sigma>2\), the kernel is hyper-singular, which should be regularized by introducing surface curl and integration by parts (Steinbach 2007).

For 3D Laplace problem, the kernel for the single layer potential is

\[U^{*}(x,y)=\frac{1}{4\pi\norm{x-y}}.\]

By letting \(t=0\), \(a=0\), \(\sigma=1\), \(\alpha=(0,0,0)\), \(S_{\alpha}(x,y)\cdot\eta_{\alpha}(\norm{y-x})=\frac{1}{4\pi}\), this kernel can be obtained by evaluating the general form. Therefore, the singularity order of the single layer potential kernel is 1.

The double layer potential kernel is

\[K(x,y)=\gamma_{1,y}^{\rm int}U^{*}(x,y)=\frac{n(y)\cdot(x-y)}{4\pi \norm{x-y}^3},\]

where \(\gamma_{1,y}^{\rm int}\) is the interior conormal derivative operator. It can be derived from the general form with \(\sigma=2\), \(t=1\), \(a=0\), \(\alpha_1=(1,0,0)\), \(\alpha_2=(0,1,0)\), \(\alpha_3=(0,0,1)\) and

\[\begin{aligned} S_{\alpha_1}(x,y)\cdot\eta_{\alpha_1}(\norm{y-x}) &= -\frac{n_1(y)}{4\pi} \\ S_{\alpha_2}(x,y)\cdot\eta_{\alpha_2}(\norm{y-x}) &= -\frac{n_2(y)}{4\pi} \\ S_{\alpha_3}(x,y)\cdot\eta_{\alpha_3}(\norm{y-x}) &= -\frac{n_3(y)}{4\pi} \\ \end{aligned}.\]

Hence, the singularity order of the double layer potential kernel is 2.

Similarly, for the adjoint double layer potential kernel

\[K^{*}(x,y)=\gamma_{1,x}^{\rm int}U^{*}(x,y)=\frac{n(x)\cdot(y-x)}{4\pi \norm{x-y}^3},\]

let

\[\begin{aligned} S_{\alpha_1}(x,y)\cdot\eta_{\alpha_1}(\norm{y-x}) &= \frac{n_1(x)}{4\pi} \\ S_{\alpha_2}(x,y)\cdot\eta_{\alpha_2}(\norm{y-x}) &= \frac{n_2(x)}{4\pi} \\ S_{\alpha_3}(x,y)\cdot\eta_{\alpha_3}(\norm{y-x}) &= \frac{n_3(x)}{4\pi} \\ \end{aligned},\]

while keeping other parameters fixed, we can see its singularity order is still 2.

For the hyper-singular potential kernel,

\[D(x,y)=-\gamma_{1,x}^{\rm int}\gamma_{1,y}^{\rm int}U^{*}(x,y)=\frac{1}{4\pi} \left[ -\frac{n(x) \cdot n(y)}{r^3} + \frac{3[n(y) \cdot (x - y)] [n(x) \cdot (x -y)]}{r^5} \right],\]

we notice that the term \([n(y) \cdot (x - y)] [n(x) \cdot (x -y)]\) can be expanded as

\[\begin{aligned} [n(y) \cdot (x - y)] [n(x) \cdot (x -y)] &= n_1(x)n_1(y)(x_1-y_1)^2 + n_2(x)n_2(y)(x_2-y_2)^2 + n_3(x)n_3(y)(x_3-y_3)^2 \\ &\quad + \left[ n_1(x)n_2(y) + n_2(x)n_1(y) \right](x_1-y_1)(x_2-y_2) \\ &\quad + \left[ n_1(x)n_3(y) + n_3(x)n_1(y) \right](x_1-y_1)(x_3-y_3) \\ &\quad + \left[ n_2(x)n_3(y) + n_3(x)n_2(y) \right](x_2-y_2)(x_3-y_3). \end{aligned}\]

This prompts us to define \(\alpha_1=(2,0,0)\), \(\alpha_2=(0,2,0)\), \(\alpha_3=(0,0,2)\), \(\alpha_4=(1,1,0)\), \(\alpha_5=(1,0,1)\), \(\alpha_6=(0,1,1)\) and

\[\begin{aligned} S_{\alpha_1}(x,y)\cdot\eta_{\alpha_1}(\norm{y-x}) &= -\frac{n(x)\cdot n(y)}{4\pi} + \frac{3n_1(x)n_1(y)}{4\pi}\\ S_{\alpha_2}(x,y)\cdot\eta_{\alpha_2}(\norm{y-x}) &= -\frac{n(x)\cdot n(y)}{4\pi} + \frac{3n_2(x)n_2(y)}{4\pi}\\ S_{\alpha_3}(x,y)\cdot\eta_{\alpha_3}(\norm{y-x}) &= -\frac{n(x)\cdot n(y)}{4\pi} + \frac{3n_3(x)n_3(y)}{4\pi}\\ S_{\alpha_4}(x,y)\cdot\eta_{\alpha_4}(\norm{y-x}) &= \frac{3[n_1(x)n_2(y)+n_2(x)n_1(y)]}{4\pi}\\ S_{\alpha_5}(x,y)\cdot\eta_{\alpha_5}(\norm{y-x}) &= \frac{3[n_1(x)n_3(y)+n_3(x)n_1(y)]}{4\pi}\\ S_{\alpha_6}(x,y)\cdot\eta_{\alpha_6}(\norm{y-x}) &= \frac{3[n_2(x)n_3(y)+n_3(x)n_2(y)]}{4\pi}\\ \end{aligned}.\]

Then the kernel function can be derived from the general form with \(\sigma=3\), \(t=2\), \(a=0\). Hence, the singularity order of the hyper-singular potential kernel is 3.

References

Erichsen, Stefan, and Stefan A. Sauter. 1998. “Efficient Automatic Quadrature in 3-d Galerkin Bem.” Computer Methods in Applied Mechanics and Engineering, Papers presented at the seventh conference on numerical methods and computational mechanics in science and engineering, 157 (3): 215–24. https://doi.org/10.1016/S0045-7825(97)00236-3.

Steinbach, Olaf. 2007. Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer Science & Business Media.