4 research outputs found
FMM-accelerated solvers for the Laplace-Beltrami problem on complex surfaces in three dimensions
The Laplace-Beltrami problem on closed surfaces embedded in three dimensions
arises in many areas of physics, including molecular dynamics (surface
diffusion), electromagnetics (harmonic vector fields), and fluid dynamics
(vesicle deformation). Using classical potential theory,the Laplace-Beltrami
operator can be pre-/post-conditioned with integral operators whose kernel is
translation invariant, resulting in well-conditioned Fredholm integral
equations of the second-kind. These equations have the standard Laplace kernel
from potential theory, and therefore the equations can be solved rapidly and
accurately using a combination of fast multipole methods (FMMs) and high-order
quadrature corrections. In this work we detail such a scheme, presenting two
alternative integral formulations of the Laplace-Beltrami problem, each of
whose solution can be obtained via FMM acceleration. We then present several
applications of the solvers, focusing on the computation of what are known as
harmonic vector fields, relevant for many applications in electromagnetics. A
battery of numerical results are presented for each application, detailing the
performance of the solver in various geometries.Comment: 18 pages, 5 tables, 3 figure
A parametrix method for elliptic surface PDEs
Elliptic problems along smooth surfaces embedded in three dimensions occur in
thin-membrane mechanics, electromagnetics (harmonic vector fields), and
computational geometry. In this work, we present a parametrix-based integral
equation method applicable to several forms of variable coefficient surface
elliptic problems. Via the use of an approximate Green's function, the surface
PDEs are transformed into well-conditioned integral equations. We demonstrate
high-order numerical examples of this method applied to problems on general
surfaces using a variant of the fast multipole method based on smooth
interpolation properties of the kernel. Lastly, we discuss extensions of the
method to surfaces with boundaries