ITERATIVE LINE CUBIC SPLINE COLLOCATION METHODS FOR ELLIPTIC PARTIAL-DIFFERENTIAL EQUATIONS IN SEVERAL DIMENSIONS

This paper presents a new class of second- and fourth-order line cubic spline collocation methods (the LCSC methods) for multidimensional li near elliptic partial differential equations with no cross derivative terms. The LCSC methods approximate the differential operator along li nes in each dimension independently and then combine the results into one large linear system. Expressed in terms of discretization stencils for the operator, these methods have nonzero entries only in the coor dinate directions. The advantage of this approach is that the discreti zation is much simpler to derive and analyze. Further, iterative metho ds are easily applied to the resulting linear systems, especially on p arallel computers. The disadvantage is that the resulting linear syste m is k times larger in k dimensions. Using the simplicity of the metho ds, iterative schemes are analyzed and formulated in order to solve th e resulting LCSC linear systems in the case of Hehmholtz problems. Blo ck Jacobi, extrapolated Jacobi (EJ), and successive overrelaxation (SO R) iteration methods are analyzed with the rates of convergence and th e optimum relaxation parameters determined. The simple structure of th e linear system makes these methods particularly suitable for parallel computation. It is shown that the overall efficiency of the method is attractive in spite of involving such a large linear system. Experime ntal results presented here confirm the convergence results for both t he discretization and iterative methods, and indicate that the converg ence results hold for problems more general than Helmholtz problems.