  This paper builds on the algebraic theory in the companion paper [Algebraic
Error Analysis for Mixed-Precision Multigrid Solvers] to obtain
discretization-error-accurate solutions for linear elliptic partial
differential equations (PDEs) by mixed-precision multigrid solvers. It is often
assumed that the achievable accuracy is limited by discretization or algebraic
errors. On the contrary, we show that the quantization error incurred by simply
storing the matrix in any fixed precision quickly begins to dominate the total
error as the discretization is refined. We extend the existing theory to
account for these quantization errors and use the resulting bounds to guide the
choice of four different precision levels in order to balance quantization,
algebraic, and discretization errors in the progressive-precision scheme
proposed in the companion paper. A remarkable result is that while iterative
refinement is susceptible to quantization errors during the residual and update
computation, the V-cycle used to compute the correction in each iteration is
much more resilient, and continues to work if the system matrices in the
hierarchy become indefinite due to quantization. As a result, the V-cycle only
requires relatively few bits of precision per level. Based on our findings, we
outline a simple way to implement a progressive precision FMG solver with
minimal overhead, and demonstrate as an example that the one dimensional
biharmonic equation can be solved reliably to any desired accuracy using just a
few V-cycles when the underlying smoother works well. In the process we also
confirm many of the theoretical results numerically.
