Numerical solution of non-separable elliptic equations by the iterative application of FFT methods

Abstract

A method for the numerical solution of non-separable (self-adjoint) elliptic equations is described in which the basic approach is the iterative application of direct methods. Such equations may be transformed into Helmholtz form and this Helmholtz problem is solved by the iterative application of FFT methods. An equation which is ‘near’ (in some sense) to the Helmholtz equation is appropriately chosen from the general class of equations soluble directly by FFT methods (see, for example, Le Bail, 1972) and the iteration (of block-Jacobi form) consists of corrections to the relevant Fourier harmonic amplitudes of the solution of this ‘nearby’ equation. It is also shown that the method is equivalent to a D' Yakonov-Gunn iteration [D' Yakonov (1961), Gunn(1964)] with a particular choice of iteration parameter and it is well known that, for self-adjoint problems with smooth coefficients, this form of iteration has a convergence rate which is essentially independent of grid-size.

In the Concus and Golub (1973) method for solving non-separable elliptic equations such equations are transformed to Helmholtz form and Poisson's equation is employed as the ‘nearby’ equation. At each iterative stage this equation is solved by the Bunemann (cyclic reduction) algorithm. Their method also uses shifted iterations to improve convergence rates and we adopt a similar approach in the present study. However, our choice of nearby equation allows the use of a form of variable shift (in addition to the constant shift used by Concus et al.,) and it will be demonstrated that the use of such a variable shift can considerably improve rates of convergence in some examples.

Numerical results are presented for illustrative examples in the unit square with Dirichlet boundary conditions and the general computational behaviour of the method was found to be in very good agreement with that predicted by a theoretical analysis. The general iterative approach may be extended to more general linear elliptic equations.

Keywords:

• Non-separable
• elliptic
• FFT
• Iterative method