On factored discretizations of the laplacian for the fast solution of Poisson's equation on general regions

A family of factored complex discretizations of the Laplacian is proposed which serve as a basis for fast, second-order accurate Poisson solvers on general two-dimensional regions. One direct implementation of these discretizations is given: a variant of the marching method which is much more stable than the latter, and thus is applicable to grids with relatively large numbers of discretization steps in each direction, without resorting to domain decomposition and multiple shooting. The gain in stability is due to solving initial value problems for two first-order difference equations, rather than one second-order equation as in the conventional marching method. These initial value problems can be interpreted as backsolves in a sparse Choleski decomposition of the coefficient matrix, induced by the factored discretization operator. The stability and accuracy of the method can be further controlled by the choice of the parameter on which the discretizations depend. The marching algorithms were tested successfully for various geometries. The factored discretization also lend themselves well to an iterative implementation by the preconditioned conjugate gradient method. © 1984 BIT Foundations.