Advanced Methods for Steady Problems—Direct Elliptic Solvers

The iterative methods for elliptic equations described in the preceding section have many attractive qualities. They require a minimal amount of computer storage, they almost always converge to the true solution to the partial differential equation and they are simple to implement in most situations. They have the disadvantage, however, that they can be very expensive, since they usually require a large number of iterations. There is another class of solvers, the direct methods, that treat the finite difference equations as a large linear system and employ some tools from linear algebra to operate on the matrix of the finite difference coefficients. These methods take advantage of the sparseness and regular structure of the coefficient matrices to minimize storage requirements and operation counts. Although they usually require more storage than the iterative methods, they require fewer operations. We describe four direct methods that can be generally divided into two categories. The first three methods seek to reduce the order of the system of equations by appropriate transformations, while the fourth method performs a lower-upper decomposition on the entire system of equations