Fast integral equation methods for the modified Helmholtz equation

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

A SVD accelerated kernel-independent fast multipole method and its application to BEM

The kernel-independent fast multipole method (KIFMM) proposed in [1] is of almost linear complexity. In the original KIFMM the time-consuming M2L translations are accelerated by FFT. However, when more equivalent points are used to achieve higher accuracy, the efficiency of the FFT approach tends to be lower because more auxiliary volume grid points have to be added. In this paper, all the translations of the KIFMM are accelerated by using the singular value decomposition (SVD) based on the low-rank property of the translating matrices. The acceleration of M2L is realized by first transforming the associated translating matrices into more compact form, and then using low-rank approximations. By using the transform matrices for M2L, the orders of the translating matrices in upward and downward passes are also reduced. The improved KIFMM is then applied to accelerate BEM. The performance of the proposed algorithms are demonstrated by three examples. Numerical results show that, compared with the original KIFMM, the present method can reduce about 40% of the iterating time and 25% of the memory requirement.Comment: 19 pages, 4 figure

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Scalable Simulation of Realistic Volume Fraction Red Blood Cell Flows through Vascular Networks

High-resolution blood flow simulations have potential for developing better understanding biophysical phenomena at the microscale, such as vasodilation, vasoconstriction and overall vascular resistance. To this end, we present a scalable platform for the simulation of red blood cell (RBC) flows through complex capillaries by modeling the physical system as a viscous fluid with immersed deformable particles. We describe a parallel boundary integral equation solver for general elliptic partial differential equations, which we apply to Stokes flow through blood vessels. We also detail a parallel collision avoiding algorithm to ensure RBCs and the blood vessel remain contact-free. We have scaled our code on Stampede2 at the Texas Advanced Computing Center up to 34,816 cores. Our largest simulation enforces a contact-free state between four billion surface elements and solves for three billion degrees of freedom on one million RBCs and a blood vessel composed from two million patches