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Stopping criteria for iterative methods: applications to PDE’s. (English) Zbl 1072.65045
The authors focus on the fact that, when considering Galerkin-type discretizations of partial differential equations (PDE), the residual $$\rho^{(n)}=A x^{(n)}-b$$ is the discrete counterpart of a linear functional, $$R^{(n)}$$, which belongs to the dual of the space that contains the exact solution. They show the difference between the exact algebraic and the functional convergence of a Krylow-based method when this method is applied to a linear system which comes from the Galerkin discretization of an elliptic PDE. They present the advantages of measuring the residual in the correct norm. Examples are presented which come from the finite element discretization of elliptic PDE’s. It is shown that measuring the residual in $$H^{-1}(\Omega)$$ gives a true evaluation of the error in the solution, whereas measuring the residual with an algebraic norm can give misleading information about the convergence.

MSC:
 65F10 Iterative numerical methods for linear systems 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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