Preconditioning cubic spline collocation discretizations of elliptic equations

Summary.

This work considers the uniformly elliptic operator $A$ defined by$Au := \ -\Delta u + a_1 u_x + a_2 u_y + a_0 u$ in $\Omega$ (the unit square) with boundary conditions:$u \ = \ 0$ on $\Gamma_0$ and${{\partial u} \over {\partial \nu }} = \alpha u$ on $\Gamma_1$ and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix$\hat A_N (h = {1 \over N} )$ . We discuss the$H^1$ condition numbers and the distribution of$\tilde \beta_N$ -singular values of the preconditioned matrices$(\tilde \beta_N )^{-1} \hat A_N$ where $\tilde \beta_N$ is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator$B$ given by$B v := - \Delta v + b_0 v$ in$\Omega$ with boundary conditions:$v \ = \ 0$ on$\Gamma_0 , \ {{\partial v } \over {\partial \nu}} \ = \ \beta v$ on $\Gamma_1$. The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by Gauss points or the space of continuous functions which are linear on the triangles of the triangulation of$\Omega$ using the Gauss points. When $A \ = \ B$ we obtain results on the eigenvalues of$\tilde \beta_N^{-1} \hat A_N$ . In the general case we obtain bounds and clustering results on the$\tilde \beta_N$ -singular values of$\tilde \beta_N^{-1} \hat A_N$ . These results are related to the results of Manteuffel and Parter [MP], Parter and Wong [PW], and Wong [W] for finite element discretizations as well as the results of Parter and Rothman [PR] for discretizations based on Legendre Spectral Collocation.