On the Construction of Discretizations of Elliptic Partial Differential Equations

Dedicated to Gerry Ladas on the occasion of his 60th Birthday Algorithmic aspects of a class of nite element collocation methods for the approximate numerical solution of elliptic partial diierential equations are described. Locally, for each nite element, the approximate solution is a polynomial. Polynomials corresponding to adjacent nite elements need not match continuously, but their values and normal derivatives match at a discrete set of points on the common boundary. High order accuracy can be attained by increasing the number of matching points and the number of collocation points for each nite element. For linear equations the collocation methods can be equivalently deened as generalized nite diierence methods. The linear (or linearized) equations that arise from the discretization lend themselves well to solution by the method of nested dissection. An implementation is described and some numerical results are given.