Fast integral equation methods for the modified Helmholtz equation

Fast integral equation methods for Rothe’s method applied to the isotropic heat equation

“…All timings cited are for a single processor on a Mac Pro 2.1 with two 3 GHz Quad-Core Intel Xeon processors. (8) is carefully analyzed in [5]. In this example, we demonstrate some of the key features.…”

“…In [5], we derive integral equation formulations to solve (5) in bounded and unbounded multiply-connected domains for both Neumann and Dirichlet boundary data. Here, we summarize the results for the bounded case with Dirichlet boundary conditions.…”

“…This evaluation can be reduced to O(N Γ ), again using the FMM (cf. [5] for more discussion on the details of implementation). Since the number of iterations needed to solve a Fredholm equation of the second kind to a fixed precision ϵ is bounded independent of N Γ , we can estimate the total cost of solving (10) by…”

“…An early example is the work by Greenbaum et al in [1] for Laplace's equation in multiply-connected domains. Subsequently, similar tools were developed for Poisson's equation [2], the Stokes equations [3], and the modified Helmholtz equation [4,5]. All of these approaches are characterized by the following: well-conditioned integral equations, if needed, are formulated, integrals are discretized using high-order quadrature, and the resulting linear systems are solved or evaluated using a fast multipole accelerated iterative solver.…”

“…In [5], we present fast, well-conditioned integral equation methods for solving the homogeneous equation in unbounded and bounded multiply-connected domains, with either Dirichlet or Neumann boundary conditions. In this paper, we present a preliminary study on coupling the methods discussed in [4] to those discussed in [5] in order to solve nonlinear diffusion-type problems in complicated domains. In order to simplify our focus, we consider problems in bounded multiply-connected domains with constant Dirichlet boundary conditions.…”