  Two essential quantities for the analysis of approximation schemes of
evolution equations are stability and convergence. We derive stability and
convergence of fully discrete approximation schemes of solutions to linear
parabolic evolution equations governed by time dependent coercive operators. We
consider abstract Galerkin approximations in space combined with theta-schemes
in time. The level of generality of our analysis comprises both a large class
of time-dependent operators and a large choice of approximating Galerkin
spaces. In particular the results apply to partial integro differential
equations for option pricing in time-inhomogeneous L\'evy models and allows for
a large variety of option types and models. The derivation builds on the strong
foundation laid out by von Petersdorff and Schwab (2003) who provide the
respective results for the time-homogeneous case. We discuss the assumptions in
the context of option pricing.
