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STOPPING CRITERIA FOR ADAPTIVE FINITE ELEMENT SOLVERS
M Arioli (STFC Rutherford Appleton Lab.)
D Loghin (Birmingham U.)
EH Georgoulis (Leicester U.)
We address the question of convergence of practical adaptive inite element solvers for symmetric elliptic problems, in the case where the resulting linear systems on each level of the algorithm are solved approximately. In particular, we show that if a particular smallness criterion (involving residuals of the linear solver and the a-posteriori bounds used in the adaptive finite element algorithm) is satisfied, then the adaptive algorithm satisfies a contraction property between two consecutive levels of refinement. This generalises the current convergence analyses of adaptive algorithms for elliptic problems whereby the resulting linear systems are assumed to be solved exactly. Moreover, based on known and new results for the estimation of the residual of the conjugate gradient method, we show that the smallness criterion gives rise to a practical stopping criterion for the iterations of the linear solver, which guarantees that the (inexact) adaptive algorithm converges. A series of numerical experiments highlights the practicality of the theoretical developments.
Adaptative finite-element method
Conjugate gradient method
SIAM J Sci Comput
SIAM J. SCI. COMPUT.
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