The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.

翻译：多重网格V循环方法是求解线性方程组的常用方法。该方法通过在细网格层进行光滑处理并在最粗网格层求解线性方程组来计算近似解。最粗网格层的求解取决于问题的规模和难度。若规模允许，通常采用基于LU或Cholesky分解的直接法。在面临大规模粗网格层问题时，常使用迭代Krylov子空间方法或基于低秩近似的直接法等近似求解器。粗网格层求解器的精度通常取决于用户针对具体问题和方法所积累的经验。本文提出一种方法，用于分析对称正定问题中粗网格层近似求解对V循环方法收敛性的影响。基于这些结果，我们推导出粗网格层停止准则，通过该准则可控制采用近似粗网格求解器的V循环方法与精确求解粗网格层问题的V循环方法所得近似解之间的差异。通过合理设定粗网格层停止准则，可使V循环方法在（几乎）相同的V循环迭代次数内收敛到选定的细网格精度水平，这与采用精确粗网格求解器的V循环方法表现一致。此外，我们利用理论结果讨论了基于相对残差范数的粗网格层停止准则中容差选择对V循环方法收敛性的影响。