Corrected Trapezoidal Rules for Near-Singular Surface Integrals Applied to 3D Interfacial Stokes Flow

Abstract

Interfacial Stokes flow can be efficiently computed using the Boundary Integral Equation method. In 3D, the fluid velocity at a target point is given by a 2D surface integral over all interfaces, thus reducing the dimension of the problem. A core challenge is that for target points near, but not on, an interface, the surface integral is near-singular and standard quadratures lose accuracy. This paper presents a method to accurately compute the near-singular integrals arising in elliptic boundary value problems in 3D. It is based on a local series approximation of the integrand about a base point on the surface, obtained by orthogonal projection of the target point onto the surface. The elementary functions in the resulting series approximation can be integrated to high accuracy in a neighborhood of the base point using a recursive algorithm. The remaining integral is evaluated numerically using a standard quadrature rule, chosen here to be the 4th order Trapezoidal rule. The method is reduced to the standard quadrature plus a correction, and is uniformly of 4th order. The method is applied to resolve Stokes flow past several ellipsoidal rigid bodies. We compare the error in the velocity near the bodies, and in the time and displacement of particles traveling around the bodies, computed with and without the corrections.

Figures (16)

• Figure 1: Stokes flow (a) past sphere, (b) past ellipsoid, (c) past two spheres, (d) past three ellipsoids.
• Figure 2: Sample set of two grids for a standard ellipsoid with $a=3$, $b=2$, $c=1$, and sample target point $\mathbf{x}_0$. (a) Grid 1, with $n_1=10$, $m_1=40$, poles on $z$-axis. (b) Grid 2, with $n_2=20$, $m_2=30$, poles on $x$-axis. For given $\mathbf{x}_0$, choose the grid whose poles are furthest from $\mathbf{x}_0$.
• Figure 3: Sketch showing surface $\mathbf{x}({\alpha},{\beta})$, discretization by $x({\alpha}_j,{\beta}_k)$, target point $\mathbf{x}_0$, and corresponding distance $d$ and base point $\mathbf{x}_b$.
• Figure 4: Error in the numerical integration of $H_{000}=1/\sqrt{x^2+y^2+d^2}$ with $d=0.001$ over the complement $W_C=D\setminus W$, where $W=[-n_w h,n_w h]\times [-n_w h,n_w h]$, $D=[-10,10]\times [-10,10]$ and $h=0.025$, as a function of $n_w$, using the 2nd, 4th and 6th order Trapezoid rules $T^2, T^4, T^6$, as indicated.
• Figure 5: Maximal error in $D[\mathbf{f}](\mathbf{x}_0)$ with $\mathbf{f}=(1,0,0)$ and $\mathbf{x}_0$ inside sphere at distance $d$ from the surface, vs $d$. The integral is computed with the uncorrected trapezoid rule $T^4[G]$ (dashed curves) and after adding the correction $E^6[H]$ (solid curves), with several values of $h = 2\pi/n$, $n$ as indicated. (a) Corrected results show loss of accuracy due to subtraction of like numbers when $d \ll 1$. (b) Corrected results are $O(h^4)$, uniformly in $d$, when the punctured trapezoidal rule is used.