Finite difference preconditioning cubic spline collocation method of elliptic equations

Abstract

We discuss a finite difference preconditioner for the \(C^1\) interpolatory cubic spline collocation method for a uniformly elliptic operator \(A\) defined by \(Au := \ -\Delta u + a_0 u\) in \(\Omega\) (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix \(\hat L_{N^2} ^{-1} \hat A_{N^2}\) where \(\hat A_{N^2}\) is the matrix of the collocation discretization operator \(A_{N^2}\) corresponding to \(A\), and \(\hat L_{N^2}\) is the matrix of the finite difference operator \(L_{N^2}\) corresponding to the uniformly elliptic operator \(L\) given by \(L v := - \Delta v + v\) in \(\Omega\) with homogeneous Dirichlet boundary conditions. Finally we mention a bound of \(H^1\)-singular values of \(\hat L^{-1}_{N^2} \hat A_{N^2}\) for a general elliptic operator \(Au := \ -\Delta u + a_1 u_x + a_2 u_y + a_0 u\) in \(\Omega\).