  Local Fourier analysis is a strong and well-established tool for analyzing
the convergence of numerical methods for partial differential equations. The
key idea of local Fourier analysis is to represent the occurring functions in
terms of a Fourier series and to use this representation to study certain
properties of the particular numerical method, like the convergence rate or an
error estimate.
  In the process of applying a local Fourier analysis, it is typically
necessary to determine the supremum of a more or less complicated term with
respect to all frequencies and, potentially, other variables. The problem of
computing such a supremum can be rewritten as a quantifier elimination problem,
which can be solved with cylindrical algebraic decomposition, a well-known tool
from symbolic computation.
  The combination of local Fourier analysis and cylindrical algebraic
decomposition is a machinery that can be applied to a wide class of problems.
In the present paper, we will discuss two examples. The first example is to
compute the convergence rate of a multigrid method. As second example we will
see that the machinery can also be used to do something rather different: We
will compare approximation error estimates for different kinds of
discretizations.
