A modified dual-level algorithm for large-scale three-dimensional Laplace and Helmholtz equation

Abstract

A modified dual-level algorithm is proposed in the article. By the help of the dual level structure, the fully-populated interpolation matrix on the fine level is transformed to a local supported sparse matrix to solve the highly ill-conditioning and excessive storage requirement resulting from fully-populated interpolation matrix. The kernel-independent fast multipole method is adopted to expediting the solving process of the linear equations on the coarse level. Numerical experiments up to 2-million fine-level nodes have successfully been achieved. It is noted that the proposed algorithm merely needs to place 2–3 coarse-level nodes in each wavelength per direction to obtain the reasonable solution, which almost down to the minimum requirement allowed by the Shannon’s sampling theorem. In the real human head model example, it is observed that the proposed algorithm can simulate well computationally very challenging exterior high-frequency harmonic acoustic wave propagation up to 20,000 Hz.

Appendices

Appendix A: The derivation of subtraction and adding-back technique

The null-fields of the boundary integral equation of the Laplace equation [44] is expressed as

$$\begin{aligned} 0=\int \limits _\Gamma {\left[ {G_0 (x,s)q(s)-\frac{\partial G_0 (x,s)}{\partial n^{e}(s)}\varphi (s)} \right] } d\Gamma (s),\quad \forall x\in \Omega ^{e},\nonumber \\ \end{aligned}$$

(A1)

where the superscript e means the exterior domain, $G_0$ is the fundamental solution of 3-D Laplace equation,

$$\begin{aligned} G_0 (x,s)=\frac{1}{r}. \end{aligned}$$

(A2)

To derive the non-singular expression of $G_0 (x_i ,s_i )$ when $x_i =s_j$, the following general solutions $\varphi (s)$ and q(s) of the 3-D Laplace equation [32, 45] are taken

$$\begin{aligned}&f(r)=\frac{1}{2}r^{2},r=\left| {s_j -x_i } \right| , \end{aligned}$$

(A3)

$$\begin{aligned}&\varphi (s_j )=\frac{\partial f(s_j -x_i )}{\partial n^{e}(x_i )}=\left\langle {(s_j -x_i )\cdot n^{e}(x_i )} \right\rangle , \end{aligned}$$

(A4)

$$\begin{aligned}&q(s_j )=\frac{\partial \varphi (s_j )}{\partial n^{e}(s_j )}=\left\langle {n^{e}(x_i )\cdot n^{e}(s_j )} \right\rangle . \end{aligned}$$

(A5)

It is obvious that when $x_i =s_j$, $\varphi (s_i )=0$ and $q(s_i )=1$. Substituting $\varphi (s_j )$ and $q(s_j )$ into Eq. (A1). We obtain

$$\begin{aligned}&\sum _{j=1}^N \left[ G_0 (x_i ,s_j )\left\langle {n^{e}(x_i )\cdot n^{e}(s_j )} \right\rangle \right. \nonumber \\&\left. \quad -\frac{\partial G_0 (x_i ,s_j )}{\partial n^{e}(s_j )}\left\langle {(s_j -x_i )\cdot n^{e}(x_i )} \right\rangle \right] A_j =0,\quad \forall x_i \in \Gamma .\nonumber \\ \end{aligned}$$

(A6)

With the collocation point x approaching the boundary, we get

$$\begin{aligned}&G_0 (x_i ,s_i )=-\frac{1}{A_i }\sum _{j=1\ne i}^N {\left[ {\begin{array}{l} G_0 (x_i ,s_j )\left\langle {n^{e}(x_i )\cdot n^{e}(s_j )} \right\rangle \\ -\frac{\partial G_0 (x_i ,s_j )}{\partial n^{e}(s_j )}\left\langle {(s_j -x_i )\cdot n^{e}(x_i )} \right\rangle \\ \end{array}} \right] }\nonumber \\&\quad A_j ,\quad \forall x_i \in \Gamma . \end{aligned}$$

(A7)

Considering that

$$\begin{aligned} \mathop {\lim }\limits _{r\rightarrow 0} \frac{e^{ikr}}{r}=\mathop {\lim }\limits _{r\rightarrow 0} \frac{\cos (kr)}{r}+\frac{\sin (kr)}{r}i=\mathop {\lim }\limits _{r\rightarrow 0} \frac{1}{r}+ki, \end{aligned}$$

(A8)

it is noted that the fundamental solutions of the 3-D Laplace equation and Helmholtz equation have the same order of singularity at origin [35, 46]. We hereby get

$$\begin{aligned} G(x_i ,s_i )=G_0 (x_i ,s_i )+ki,\quad r\rightarrow 0, \end{aligned}$$

(A9)

where $G={e^{ikr}}/r$ is the fundamental solution of 3-D Helmholtz equation.

Using Eqs. (A7) and (A9), the singularities of the basis function is avoided.

Appendix B: The pseudocode of the DL-SBM