Additive Schwarz method: Difference between revisions

Revision as of 16:28, 17 June 2011

Revision as of 17:52, 16 November 2011

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=== Solving on a computer ===

A typical way of doing this is to ''sample'' ''f'' at regular [[interval (mathematics)|intervals]]s in the [[Square (geometry)|square]] [0,1] × [0,1]. For instance, we could take 8 samples in the ''x'' direction at ''x'' = 0.1, 0.2, ..., 0.8 and 0.9, and 8 samples in the ''y'' direction at similar [[coordinate system|coordinates]]. We would then have 64 samples of the square, at places like (0.2,0.8) and (0.6,0.6). The goal of the [[computer program]] would be to calculate the value of ''f'' at those 64 points, which seems easier than finding an abstract function of the square.

There are some difficulties, for instance it is not possible to calculate ''f''_''xx''(0.5,0.5) knowing ''f'' at only 64 points in the square. To overcome this, one uses some sort of numerical approximation of the derivatives, see for instance the [[finite element method]] or [[finite difference]]s. We ignore these difficulties and concentrate on another aspect of the problem.