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A short survey on preconditioning techniques in spectral calculations. (English) Zbl 0964.65132
Summary: Two decades ago collocation methods for high accuracy solutions of partial differential equations seemed applicable only to special classes of problems with simple geometries, regular solutions, absence of numerical round-off errors, etc. The usefulness of these methods in engineering problems was very much in question. The situation has changed with many obstacles having been removed which prevented spectral methods to be used with maximum performance. We briefly review a key element in this change: the use of preconditioning techniques.

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
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