Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

Abstract

We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all distances to the domain boundary as well as on the boundary itself. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon quadrature by expansion, which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients to match the potential on a smaller concentric ring. We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.

Appendix A: List of kernels

Here we list the kernels for the single- and double-layer potentials for the PDEs considered text. In each case ${\varvec{x}}$ and ${\varvec{y}}$ are points in $\mathbb {R}^2$ and ${\varvec{r}}\mathrel {\mathop :}={\varvec{x}}-{\varvec{y}}$. The single-layer kernel is the fundamental solution. In double-layer kernels, ${\varvec{n}}$ is the unit vector denoting the dipole direction, which in the context of boundary integral formulation is the outward pointing normal to the surface.

• Laplace:

  $$\begin{aligned} \varDelta u&= 0, \end{aligned}$$

  (A.1)
  $$\begin{aligned} S({\varvec{x}},{\varvec{y}})&= -\frac{1}{2\pi }\log |{\varvec{r}}|,\end{aligned}$$

  (A.2)
  $$\begin{aligned} D({\varvec{x}},{\varvec{y}})&= \frac{1}{2\pi }\frac{{\varvec{r}}\cdot {\varvec{n}}}{|{\varvec{r}}|^2}, \end{aligned}$$

  (A.3)
  $$\begin{aligned} \lim _{{\varvec{y}}\rightarrow {\varvec{x}}} D({\varvec{x}},{\varvec{y}})&= -\frac{\kappa }{4\pi }, \qquad {\varvec{x}},{\varvec{y}}\in \varGamma , \quad \text {(where }\kappa \text { is the signed curvature)}. \end{aligned}$$

  (A.4)

• Yukawa:

  $$\begin{aligned} \varDelta u- \lambda ^2 u&= 0, \end{aligned}$$

  (A.5)
  $$\begin{aligned} S({\varvec{x}},{\varvec{y}})&= \frac{1}{2\pi }K_0(\lambda |{\varvec{r}}|), \end{aligned}$$

  (A.6)
  $$\begin{aligned} D({\varvec{x}},{\varvec{y}})&= \frac{\lambda }{2\pi }\frac{{\varvec{r}}\cdot {\varvec{n}}}{|{\varvec{r}}|}K_1(\lambda |{\varvec{r}}|), \end{aligned}$$

  (A.7)

  where $K_0, K_1$ are modified Bessel functions of the second kind of order zero and one, respectively.

• Helmholtz:

  $$\begin{aligned} \varDelta u+ \omega ^2 u&= 0, \end{aligned}$$

  (A.8)
  $$\begin{aligned} S({\varvec{x}},{\varvec{y}})&= \frac{i}{4}H_0^1(\omega |{\varvec{r}}|), \end{aligned}$$

  (A.9)
  $$\begin{aligned} D({\varvec{x}},{\varvec{y}})&= \frac{i\omega }{4}\frac{{\varvec{r}}\cdot {\varvec{n}}}{|{\varvec{r}}|} H_1^1(\omega |{\varvec{r}}|), \end{aligned}$$

  (A.10)

  where $H^1_0, H^1_1$ are respectively modified Hankel functions of the first kind of order zero and one.

• Stokes:

  $$\begin{aligned} -\varDelta {\varvec{u}}+ \nabla p&= 0, \qquad {{\mathrm{\nabla \cdot }}}{\varvec{u}}= 0, \end{aligned}$$

  (A.11)
  $$\begin{aligned} S({\varvec{x}},{\varvec{y}})&= \frac{1}{4\pi }\left( -\log |{\varvec{r}}| + \frac{{\varvec{r}}\otimes {\varvec{r}}}{|{\varvec{r}}|^2}\right) , \end{aligned}$$

  (A.12)
  $$\begin{aligned} D({\varvec{x}},{\varvec{y}})&= \frac{{\varvec{r}}\cdot {\varvec{n}}}{\pi }\frac{{\varvec{r}}\otimes {\varvec{r}}}{|{\varvec{r}}|^4}, \end{aligned}$$

  (A.13)
  $$\begin{aligned} \lim _{{\varvec{y}}\rightarrow {\varvec{x}}} D({\varvec{x}},{\varvec{y}})&= -\frac{\kappa }{2\pi } \varvec{t}\otimes \varvec{t}, \end{aligned}$$

  (A.14)
  $$\begin{aligned} P({\varvec{x}},{\varvec{y}})&= -\frac{1}{\pi |{\varvec{r}}|^2} \left( 1 - 2\frac{{\varvec{r}}\otimes {\varvec{r}}}{|{\varvec{r}}|^2} \right) {\varvec{n}}. \end{aligned}$$

  (A.15)

• Navier: Linear elasticity for isotropic material with shear modulus $\mu$ and Poisson ratio $\nu$,

  $$\begin{aligned}&\mu \varDelta {\varvec{u}}+ \frac{\mu }{1-2\nu } {{\mathrm{\nabla }}}{{\mathrm{\nabla \cdot }}}{\varvec{u}}= 0,\end{aligned}$$

  (A.16)
  $$\begin{aligned}&S({\varvec{x}},{\varvec{y}}) = -\frac{3-4\nu }{8\pi (1-\nu )}\log |{\varvec{r}}| + \frac{1}{8\pi (1-\nu )} \frac{{\varvec{r}}\otimes {\varvec{r}}}{|{\varvec{r}}|^2}, \end{aligned}$$

  (A.17)
  $$\begin{aligned}&D({\varvec{x}},{\varvec{y}}) = \frac{1-2\nu }{4\pi (1-\nu )}\left( \frac{{\varvec{r}}\cdot {\varvec{n}} + {\varvec{n}}\otimes {\varvec{r}}- {\varvec{r}}\otimes {\varvec{n}}}{|{\varvec{r}}|^2} + \frac{2}{1-2\nu } \frac{{\varvec{r}}\cdot {\varvec{n}}\, {\varvec{r}}\otimes {\varvec{r}}}{|{\varvec{r}}|^4}\right) . \end{aligned}$$

  (A.18)