Numerical Solution of the 3D Poisson Equation with the Method of Local Corrections

Jul
22
2014
10:00 AM - 11:00 AM
: Christos Kavouklis, Lawrence Berkeley National Laboratory, Berkeley, California
Computer Science and Mathematics Division Seminar
Research Office Building (5700), Room L-204
CONTACT :
Email: Cory Hauck
Phone:865.574.0730
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We present a new version of the Method of Local Corrections; a low communications algorithm for the numerical solution of the free space Poisson's equation on 3D structured grids. We are assuming a decomposition of the fine computational domain (which contains the global right hand side - charge) into a set of small disjoint cubic patches (e.g. of size 33^3). The Method of Local Corrections comprises three steps where Mehrstellen discretizations of the Laplace operator are employed; (i) A loop over the fine disjoint patches and the computation of local potentials on sufficiently large extensions of theirs (downward pass) (ii) An inexpensive global Poisson solve on the associated coarse domain with right hand side computed by applying the coarse mesh Laplacian to the local potentials of step (i) and (iii) A correction of the local solutions computed in step (i) on the boundaries of the fine disjoint patches based on interpolating the global coarse solution and a propagation of the corrections in the patch interiors via local Dirichlet solves (upward pass). Local solves in the downward pass and the global coarse solve are performed utilizing the domain doubling algorithm of Hockney. For the local solves in the upward pass we are employing a standard DFT Dirichlet Poisson solver. In this new version of the Method of Local Corrections we take into consideration the local potentials induced by truncated Legendre expansions of degree P of the local charges (the original version corresponded to P=0). The result is an h-p scheme that is P+1-order accurate and involves only local communication. Specifically, we only have to compute and communicate the coefficients of local Legendre expansions (that is, for instance, 20 scalars per patch for expansions of degree P=3). Several numerical simulations are presented to illustrate the new method and demonstrate its convergence properties.

If you would like to meet Chris Kavouklis, please contact Clay Webster at webstercg@ornl.gov or Liz Hebert at hebertem@ornl.gov

 

 

 

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