Cubic spline finite element method for solving Poisson's equation on a square

Abstract

We solve a simple 2-point boundary value problem with Dirichlet boundary conditions on [0,1] using a finite element method with cubic splines. We obtain explicit forms for the mass and stiffness matrices that arise from the method. We then solve Poisson's equation with zero Dirichlet boundary conditions in the unit square using a finite element method with basis functions that are tensor products of cubic splines. The resulting linear systems are solved in Matlab using Gauss elimination without pivoting, and then more efficiently using a Matrix Decomposition Algorithm. Cost of the matrix decomposition algorithm is O(N[superscript 3]), where N+1 is the number of subintervals in each coordinate direction. We improve the method still by using an Alternating Direction Implicit Method that reduces the cost of solving the resulting linear systems to O(N[superscript 2]ln[superscript 2]N).