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5. Hollow dies
This section briefly describes specific methods for dealing with large problems with few unknowns in each equation.
5.1 Large problems in linear algebra
Problems obtained by discretizing partial differential equations easily generate gigantic matrices. For example, discretization of the heat equation in dimension 3 on a parallelepiped [0, 1] 3 requires the domain to be cut along three directions; if we take as unknowns the temperature at each point of the discretized cube, we have N 3 unknowns if each side of the cube is discretized into N points. We'll also have as many (linear) equations by writing the discrete approximations of the Laplacian and partial derivatives,...
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Hollow dies
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Numerical calculation software and libraries
The actual implementation of the methods described above requires highly precise computer techniques which the size of this article does not allow us to cover. In any case, "off-the-shelf" programs are not always well-suited to real-life situations, which may require prior simplifications, estimates of tolerable errors, etc.
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