Problems from Computational Physics

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 [J4] [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.

More recently, Howard Elman, Oliver Ernst, and I have 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.

[J4]

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.