Finite difference preconditioning cubic spline collocation method of elliptic equations

We discuss a finite difference preconditioner for the C ¹ interpolatory cubic spline collocation method for a uniformly elliptic operator A defined by Au := -Δu + a₀u in Ω (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix L̂^-1_N2Â_N2 where Â_N2 is the matrix of the collocation discretization operator A_N2 corresponding to A, and L̂_N2 is the matrix of the finite difference operator L_N2 corresponding to the uniformly elliptic operator L given by Lυ := -Δυ + υ in Ω with homogeneous Dirichlet boundary conditions. Finally we mention a bound of H ¹-singular values of L̂^-1_N2Â_N2 for a general elliptic operator Au := -Δu + a₁u_x + a₂u_y + a₀u in Ω.