  We consider stopping criteria that balance algebraic and discretization
errors for the conjugate gradient algorithm applied to high-order finite
element discretizations of Poisson problems. Firstly, we introduce a new
stopping criterion that suggests stopping when the norm of the linear system
residual is less than a small fraction of an error indicator derived directly
from the residual. This indicator shares the same mesh size and polynomial
degree scaling as the norm of the residual, resulting in a robust criterion
regardless of the mesh size, the polynomial degree, and the shape regularity of
the mesh. Secondly, for solving Poisson problems with highly variable piecewise
constant coefficients, we introduce a subdomain-based criterion that recommends
stopping when the norm of the linear system residual restricted to each
subdomain is smaller than the corresponding indicator also restricted to that
subdomain. Reliability and efficiency theorems for the first criterion are
established. Numerical experiments, including tests with highly variable
piecewise constant coefficients and a GPU-accelerated three-dimensional
elliptic solver, demonstrate that the proposed criteria efficiently avoid both
premature termination and over-solving.
