Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

We derive new formulas for the fundamental solutions of slow, viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solvers libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero slip condition) using a single reflected Stokeslet combined with a single Papkovich-Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation

A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

AbstractWe present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new

The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering

We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials ${\bf A},\phi$ in the Lorenz gauge, we establish boundary conditions on the potentials themselves, rather than on the field quantities. This permits the development of a well-conditioned second kind Fredholm integral equation which has no spurious resonances, avoids low frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, $\phi^{Sc}$, is determined entirely by the incident scalar potential $\phi^{In}$. Likewise, the unknown vector potential defining the scattered field, ${\bf A}^{Sc}$, is determined entirely by the incident vector potential ${\bf A}^{In}$. This decoupled formulation is valid not only in the static limit but for arbitrary $\omega\ge 0$.Comment: 33 pages, 7 figure

Coulomb Interactions On Planar Structures: Inverting the Square Root of the Laplacian

We present an adaptive fast multipole method for inverting the square root of the Laplacian in two dimensions. Solving this problem is the dominant computational cost in many applications arising in electrical engineering, geophysical fluid dynamics, and the study of thin films. I corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity. Our algorithmis direct and assumes only that the source distribution is discretized using an adaptive quad-tree. The amount of work grows linearly with the number of mesh points