Annals of Mathematical Sciences and Applications

Volume 6 (2021)

Number 2

Stopping criteria based on a posteriori error estimations for iterative solvers of convection-diffusion equations

Pages: 173 – 195



Chin-Tien Wu (Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan)

Ping-Kong Huang (Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan)


In this work, sparse linear systems obtained from the streamline diffusion finite element discretization of the convection-diffusion equations are solved by a multigrid method and the generalized minimal residual method. Adaptive mesh refinement process is considered as an integral part of the solution process for increasing numerical accuracy on the boundary layers and internal layers of the solutions. We propose two stopping criteria for iterative solvers to ensure that the iterative errors are within the range of the a posteriori error bound. Under the assumptions (29) and (30) that the a priori error bound and the a posteriori error indicator do not change rapidly during mesh refinement processes, we show that the error indicators computed from iterative solutions satisfying the proposed stopping criteria are as reliable and efficient as the error indicators computed from direct solutions. Moreover, our numerical results show that iterative steps are reduced significantly for the multigrid solver to satisfy the proposed stopping criteria. The refined meshes obtained from such iterative solutions are almost identical with the refined meshes obtained from direct solutions.


stopping criteria, a posteriori error, convection-diffusion, GMRES, MG, SDFEM

The first-named author is supported by the Ministry of Science and Technology of Taiwan (grant no. 108-2115-M-009-002-MY2).

Received 28 April 2021

Accepted 10 September 2021

Published 18 October 2021