Iterative Methods for Linear Systems

By Chris H. Rycroft. Explorations in Numerical Analysis, 2019

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For many elliptic PDE problems, finite-difference and finite-element methods are the techniques of choice. In a finite-difference approach, we search for a solution uk on a set of discrete gridpoints 1, . . . , k. The discretized partial differential equation and boundary conditions give us linear relationships between the different values of uk. In a finite-element method, we express our solution 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 which satisfy all the linear relationships prescribed by the PDE. This can be written as a matrix equation of the form

Subfields: Computational Theory and Mathematics, Numerical Analysis

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