Solving elliptic PDEs in more than one dimension can be a computationally
expensive task. For some applications characterised by a high degree of
anisotropy in the coefficients of the elliptic operator, such that the term
with the highest derivative in one direction is much larger than the terms in
the remaining directions, the discretized elliptic operator often has a very
large condition number - taking the solution even further out of reach using
traditional methods. This paper will demonstrate a solution method for such
ill-behaved problems. The high condition number of the D-dimensional
discretized elliptic operator will be exploited to split the problem into a
series of well-behaved one and (D-1)-dimensional elliptic problems. This
solution technique can be used alone on sufficiently coarse grids, or in
conjunction with standard iterative methods, such as Conjugate Gradient, to
substantially reduce the number of iterations needed to solve the problem to a
specified accuracy. The solution is formulated analytically for a generic
anisotropic problem using arbitrary coordinates, hopefully bringing this method
into the scope of a wide variety of applications.Comment: 37 pages, 11 figure
