James W. Swift

John M. Neuberger, Nandor Sieben, James W. Swift

In this paper we numerically solve the eigenvalue problem $Δu + λu = 0$ on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved accuracy over the previous computations of Lapidus, Neuberger, Renka & Griffith. We extrapolate the results for grid spacing $h$ to the limit $h \rightarrow 0$ in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapdus et al. We analyze the symmetry of the region to explain the multiplicity-two eigenvalues, and present a canonical choice of the two eigenfunctions that span each two-dimensional eigenspace.

John M. Neuberger, Nandor Sieben, James W. Swift

We present a simple and easy to apply methodology for using high-level self-submitting parallel job queues in an MPI environment. Using C++, we implemented a library of functions, MPQueue, both for testing our concepts and for use in real applications. In particular, we have applied our ideas toward solving computational combinatorics problems and for finding bifurcation diagrams of solutions of partial differential equations (PDE). Our method is general and can be applied in many situations without a lot of programming effort. The key idea is that workers themselves can easily submit new jobs to the currently running job queue. Our applications involve complicated data structures, so we employ serialization to allow data to be effortlessly passed between nodes. Using our library, one can solve large problems in parallel without being an expert in MPI. We demonstrate our methodology and the features of the library with several example programs, and give some results from our current PDE research. We show that our techniques are efficient and effective via overhead and scaling experiments.