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A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles

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10

Spectral methods

10.1  Sparse matrices vs. small matrices

In the previous two chapters we have introduced methods based on completely different principles: finite differences rest upon the replacement of derivatives by linear combinations of function values but the idea behind finite elements is to approximate an infinite-dimensional expansion of the solution in a finite-dimensional space. Yet the implementation of either approach ultimately leads to the solution of a system of algebraic equations. The bad news about such a system is that it ...

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