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-cycle方法是求解线性方程组的常用方法。它通过在细层进行平滑处理并在最粗层求解线性方程组来计算近似解。最粗层的求解取决于问题的规模和复杂性。如果问题规模允许，通常采用基于LU分解或Cholesky分解的直接方法。在处理大规模最粗层问题时，常使用迭代Krylov子空间方法或基于低秩近似的直接方法作为近似求解器。最粗层求解器的精度通常由用户根据具体问题和方法的经验确定。本文针对对称正定问题，提出了一种分析近似最粗层求解对V-cycle方法收敛性影响的方法。基于这些结果，我们推导出最粗层停止准则，通过该准则可控制使用近似最粗层求解器的V-cycle方法计算的近似解与精确求解最粗层问题所得近似解之间的差异。因此，可设置最粗层停止准则，使得V-cycle方法在（几乎）与使用精确最粗层求解器相同的V-cycle迭代次数内，收敛到指定的最细层精度。我们还利用理论结果讨论了基于相对残差范数的最粗层停止准则中容忍度选择对V-cycle方法收敛性的影响。