On the Iterative Solution of Line Spline Collocation Schemes for Elliptic PDEs

In this paper we present the convergence analysis of iterative schemes for solving linear systems resulting from diacretizing multidimensional linear second order elliptic partial differential equations (PDEs) defined in a hyper-parallelepiped n and subject to Dirichlet boundary conditions on some faces of n and Neumann on the others, using a new class of line cubic spline collocation (LeSC) methods. These LeSe methods approximate the differential operator along lines in each dimension independently and then combine the resulting equations into one large non-symmetric linear system of equations which lacks many of the properties found in Ritz-Galerkin type finite element methods. Nevertheless, we derive analytic expressions for the spectral radius of the corresponding Jacobi iteration matrix and from this we determine the convergence ranges and compute the optimal parameters for the Extrapolated Jacobi and SOR methods. Experimental results presented confirm the theoretical convergence results and indicate that the latter hold for problems more general than Helmholtz problems.