Iterative methods for linear systems
نویسنده
چکیده
For many elliptic PDE problems, finite-difference and finite-element methods are the techniques of choice. In a finite-difference approach, a solution uk on a set of discrete gridpoints 1, . . . , k is searched for. The discretized partial differential equation and boundary conditions create linear relationships between the different values of uk. In the finite-element method, the solution is expressed as a linear combination uk of basis functions λk on the domain, and the corresponding finite-element variational problem again gives linear relationships between the different values of uk. Regardless of the precise details, all of these approaches ultimately end up with having to find the uk that satisfy all the linear relationships prescribed by the PDE. This can be written as a matrix equation of the form
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