Convergence Criteria for Iterative Methods in Solving Convection-diffusion Equations on Adaptive Meshes

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 residule method. Adaptive mesh refinement process is considered as an integral part of the solution process. We propose some stopping criteria for iterative solvers to ensure the iterative errors are within the range of the a posteriori error bound. Under the assumption that the error indicators do not change rapidly during mesh refinement processes, we also 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 directive 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 indistinguishable with the refined meshes obtained from directive solutions.