Constructive techniques for approximating collocation linear systems

A high order discretization by spectral collocation methods of the elliptic problem [GRAPHICS] is considered where A(x) = a(x)I_2, x = (x^([1]), x^([2])) and I_2 denotes the 2 x 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 {A(n)}(n) over the nonuniform grid sequence defined via the collocation points [11]. Were we propose a preconditioning strategy for {A(n)}_(n) based on the "approximate factorization" idea. More specifically, the preconditioning sequence {P-n}_(n) is constructed by using two basic structures: a FD discretization of (1) with A(x) = I_2 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 (P_n^(-1) A_(n)}_(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