Analysis of Iterative Line Spline Collocation Methods for Elliptic Partial Differential Equations

In this paper we present the convergence analysis of iterative schemes for solving linear systems resulting from discretizing multidimensional linear second-order elliptic partial differential equations (PDEs) defined in a hyperparallelepiped $\Omega$ and subject to Dirichlet boundary conditions on some faces of $\Omega$ and Neumann on the others, using line cubic spline collocation (LCSC) methods. Specifically, we derive analytic expressions or obtain sharp bounds 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 successive overrelaxation (SOR) methods. Experimental results are also presented.