Abstract
A high order discretization by spectral collocation methods of the elliptic problem
is considered where A(x)=a(x)I 2, x=(x [1],x [2]) and I 2 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 {A n } n over the nonuniform grid sequence defined via the collocation points [11]. Here 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.
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Serra-Capizzano, S., Tablino-Possio, C. Constructive techniques for approximating collocation linear systems. Numerical Algorithms 25, 323–339 (2000). https://doi.org/10.1023/A:1016621409981
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DOI: https://doi.org/10.1023/A:1016621409981