  This paper establishes the first theoretical framework for analyzing the
rounding-error effects on multigrid methods using mixed-precision
iterative-refinement solvers. While motivated by the sparse symmetric positive
definite (SPD) matrix equations that arise from discretizing linear elliptic
PDEs, the framework is purely algebraic such that it applies to matrices that
do not necessarily come from the continuum. Based on the so-called energy or
$A$ norm, which is the natural norm for many problems involving SPD matrices,
we provide a normwise forward error analysis, and introduce the notion of
progressive precision for multigrid solvers. Each level of the multigrid
hierarchy uses three different precisions that each increase with the fineness
of the level, but at different rates, thereby ensuring that the bulk of the
computation uses the lowest possible precision. The theoretical results
developed here in the energy norm differ notably from previous theory based on
the Euclidean norm in important ways. In particular, we show that simply
rounding an exact result to finite precision causes an error in the energy norm
that is proportional to the square root of $\kappa$, the associated matrix
condition number. (By contrast, this error is of order $1$ when measured in the
Euclidean norm.) Given this observation, we show that the limiting accuracy for
both V-cycles and full multigrid is optimal in the sense that it is also
proportional to $\kappa^{1/2}$ in energy. Additionally, we show that the loss
of convergence rate due to rounding grows in proportion to $\kappa^{1/2}$, but
argue that this loss is insignificant in practice. The theory presented here is
the first forward error analysis in the energy norm of iterative refinement and
the first rounding error analysis of multigrid in general.
