A Non-Iterative Numerical Solver of Poisson and Helmholtz Equations Using High-Order Finite-Element Functions

• Published in: Advances in Quantum Chemistry Vol. 50; pp. 235 - 247
• Main Authors: Berger, Raphael J.F., Sundholm, Dage
• Format: Book Chapter
• Language: English
• Published: United States Elsevier Science & Technology 2005
• Subjects: Physical chemistry; Quantum & theoretical chemistry; Quantum physics (quantum mechanics & quantum field theory)
• ISBN: 9780120348503; 0120348500
• ISSN: 0065-3276; 2162-8815
• DOI: 10.1016/S0065-3276(05)50011-X

A non-iterative finite-element solver for n-dimensional Poisson and Helmholtz equations has been developed. The electrostatic potential and the charge-density distributions are expanded in finite-element functions consisting of up to sixth-order Lagrange interpolation functions. The method can also be applied to differential equations in more than three-dimensional spaces. It is efficient and well suited for parallel computers, since the innermost loops constitute matrix multiplications and the outer ones can be used as parallelizing index on a parallel computer. The solution of the n-dimensional Poisson and Helmholtz equations scales as N(n+1n), where N=Nin denotes the grid size and Ni is the number of element functions, i.e., the number grid points, in each dimension.