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A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements. (English) Zbl 1186.65149

The author proposes a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. For the sake of simplicity, the author considers the Laplace problem in \(\mathbb R^{2}\) with homogeneous boundary conditions
\[ \begin{cases} -\Delta u = f &\text{in }\Omega,\\ u = 0 &\text{on }\partial\Omega. \end{cases} \]
Moreover, the simplest finite element method is presented, namely continuous, piecewise linear finite elements are considered. The paper is organized as follows. First, the model problem and the error estimator are introduced. Second, the equivalence between the true error and the error estimator is proved. In the third part of the paper, an adaptive finite element algorithm that includes a new stopping criterion is proposed. This algorithm is based on a posteriori error estimates of the previous section and on heuristics. Finally, numerical results are presented in the last section of the paper.
The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. The author proposes to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution \(u\) and the computed solution \(u_{h}^{n}\) (the superscript \(n\) stands for the CG iteration number, the subscript \(h\) for the typical mesh size) and on heuristics to relate the error between \(u_{h}\) and \(u_{h}^{n}\) to the residual vector. Numerical experiments with anisotropic adaptive meshes show that the total number of CG iterations can be divided by 10 without significant discrepancy in the computed results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65J05 General theory of numerical analysis in abstract spaces

Software:

BL2D-V2; ALBERT
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References:

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