Physics Applications

Helmholtz equations are used to model a variety of important physical systems, ranging from heat distribution to the transmission of sound. Olof Widlund and I developed efficient algorithms for solving the Helmholtz equation on general three dimensional regions with Dirichlet or Neumann boundary conditions, imbedding the region in a cube. Innovations involved the proof of existence of the discrete solution, development of effective scaling strategies, and the choice of effective storage structures [J04] [J10]. Efficient variants of these algorithms for problems with mixed boundary conditions over a union of rectangles were later developed [J25], with application to a National Bureau of Standards (now National Institute of Standards and Technology) model of smoke transport in buildings.

Later, Howard Elman, Oliver Ernst, and I considered the difficulties encountered when the Helmholtz parameter is negative, leading to indefinite systems of linear equations. Results are presented in [J46] [J52] [J60]. These problems arise in studying wave phenomena, for example, transmission of sound underwater. In collaboration with post-doc Michael Stewart, we extended our study to problems in which the boundary conditions are stochastic [J68].

A method for solving an important physics problem, approximating the number of monomer-dimer coverings in periodic lattices, was given in [J58]. This model has a variety of uses in solid state physics, ranging from studying spontaneous magnetization to phase transitions in multicomponent liquids and biological membranes.

The behavior of systems such as superconductors are described using Schrödinger's equation. In [J86], Michael O'Hara and I considered the effects of perturbations on the behavior of solutions to this equation.

[J04]

Dianne P. O'Leary and Olof Widlund, ``Capacitance matrix methods for the Helmholtz equation on general three dimensional regions,'' Mathematics of Computation 33 (1979) 849-879.

[J10]

Dianne P. O'Leary and Olof Widlund, ``Algorithm 572: Solution of the Helmholtz equation for the Dirichlet problem on general bounded three dimensional regions,'' ACM Transactions on Mathematical Software 7 (1981) 239-246.

[J25]

Dianne P. O'Leary, ``A note on the capacitance matrix algorithm, substructuring, and mixed or Neumann boundary conditions,'' Applied Numerical Mathematics 3 (1987) 339-345.

[J46]

Howard C. Elman and Dianne P. O'Leary ``Efficient Iterative Solution of the Three-Dimensional Helmholtz Equation," Journal of Computational Physics, 142 (1998) 163-181.

[J52]

Howard C. Elman and Dianne P. O'Leary, "Eigenanalysis of Some Preconditioned Helmholtz Problems," Numerische Mathematik, 83 (1999) 231-257.

[J58]

Isabel Beichl, Dianne P. O'Leary, and Francis Sullivan, ``Approximating the Number of Monomer-Dimer Coverings in Periodic Lattices," Physical Review E 64 (2001) 016701.1-6.

[J60]

Howard C. Elman, Oliver G. Ernst, and Dianne P. O'Leary, ``A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations," SIAM J. on Scientific Computing, 23 (2001) 1290-1314.

[J68]

Howard C. Elman, Oliver G. Ernst, Dianne P. O'Leary, and Michael Stewart, ``Efficient Iterative Algorithms for the Stochastic Finite Element Method with Application to Acoustic Scattering," Computer Methods in Applied Mechanics and Engineering, 194 (2005) 1037-1055.

[J86]

Michael J. O'Hara and Dianne P. O'Leary, ``The Adiabatic Theorem in the Presence of Noise," Physical Review A, 77 (2008) 042319, 20 pages. http://link.aps.org/abstract/PRA/v77/e042319DOI: 10.1103/PhysRevA.77.042319. Chosen for inclusion in Virtual Journal of Applications of Superconductivity 14:9 (2008) and Virtual Journal of Quantum Information (2008).