Constructive techniques for approximating collocation linear systems

Abstract

A high order discretization by spectral collocation methods of the elliptic problem $$\begin{gathered} \left\{ \begin{gathered} A_{A,b} u \equiv - \nabla [A(x)\nabla u(x)] + b(x)u(x) = c(x){\text{ on}}\;\Omega \; = ( - 1,1)^2 \hfill \\ u\left| {\partial \Omega \equiv 0,} \right. \hfill \\ \end{gathered} \right. \hfill \\ \hfill \\ \end{gathered} $$ is considered where A(x)=a(x)I2, x=(x[1],x[2]) and I2 denotes the 2×2 identity matrix, giving rise to a sequence of dense linear systems that are optimally preconditioned by using the sparse Finite Difference (FD) matrix-sequence {An}n over the nonuniform grid sequence defined via the collocation points [11]. Here we propose a preconditioning strategy for {An}n based on the “approximate factorization” idea. More specifically, the preconditioning sequence {Pn}n is constructed by using two basic structures: a FD discretization of (1) with A(x)=I2 over the collocation points, which is interpreted as a FD discretization over an equidistant grid of a suitable separable problem, and a diagonal matrix which adds the informative content expressed by the weight function a(x). The main result is the proof that the sequence {Pn−1An}n is spectrally clustered at unity so that the solution of the nonseparable problem (1) is reduced to the solution of a separable one, this being computationally more attractive [2,3]. Several numerical experiments confirm the goodness of the discussed proposal.