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A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles

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11

Gaussian elimination for sparse linear equations

11.1  Banded systems

Whether the objective is to solve the Poisson equation using finite differences, finite elements or a spectral method, the outcome of discretization is a set of linear algebraic equations, e.g. (8.16) or (9.7). The solution of such equations ultimately constitutes the lion’s share of computational expenses. This is true not just with regard to the Poisson equation or even elliptic PDEs since, as will become apparent in Chapter 16, the practical computation of parabolic PDEs ...

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