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,...
The Ultimate Scientific and Technical Reference
This article is included in
Mathematics
This offer includes:
Knowledge Base
Updated and enriched with articles validated by our scientific committees
Services
A set of exclusive tools to complement the resources
Practical Path
Operational and didactic, to guarantee the acquisition of transversal skills
Doc & Quiz
Interactive articles with quizzes, for constructive reading
Hollow dies
Find out more
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.
The following is a...
References
The Ultimate Scientific and Technical Reference
