A non-iterative numerical solver of Poisson and Helmholtz equations using high-order finite-element functions

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+1)/(n)), where N = N-i(n) denotes the grid size and N-i is the number of element functions, i.e., the number grid points, in each dimension.