A Non-Iterative Numerical Solver of Poisson and Helmholtz Equations Using High-Order Finite-Element Functions - Berger, Raphael J.F.; Sundholm, Dage (Advances in Quantum Chemistry, 2005)
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.