Mat-1.3656 Seminar on
numerical analysis and computational science
Monday Oct 20, 2008, room
U356 at 14.15, Eirola & Stenberg
Dage Sundholm, Helsingin yliopisto
Tensorial finite-element methods in quantum chemistry
A
method for calculating the electrostatic potential directly in a
straightforward manner is presented. While traditional methods for
calculating the electrostatic potential usually involve solving the
Poisson equation iteratively, we obtain the electrostatic interaction
potential by performing direct numerical integration of the Coulomb
lawexpression using tensorial finite-element functions defined on a
grid. The singularity of the Coulomb operator is circumvented by an
integral transformation and the resulting auxilary integral is obtained
using Gauss quadrature. The three-dimensional finite-element basis is
constructed as a tensor (outer) product of one-dimensional functions
yielding a partial factorization of the expessions. The resulting
algorithm has without using any prescreening or other computational
tricks a formal computational scaling of 4N/3, where N is the size of
the grid. By prescreening, the method can be made to scale linearly
with the grid size. It has been implemented for efficiently running on
parallel computers. The matrix multiplications of the innermost loops
are completely independent yielding a parallel algorithm with the
computational costs scaling practically linearly with the number of
processors.