  This paper considers a large class of linear operator equations, including
linear boundary value problems for partial differential equations, and treats
them as linear recovery problems for objects from their data. Well-posedness of
the problem means that this recovery is continuous. Discretization recovers
restricted trial objects from restricted test data, and it is well-posed or
stable, if this restricted recovery is continuous. After defining a general
framework for these notions, this paper proves that all well-posed linear
problems have stable and refinable computational discretizations with a
stability that is determined by the well-posedness of the problem and
independent of the computational discretization. The solutions of discretized
problems converge when enlarging the trial spaces, and the convergence rate is
determined by how well the full data of the object solving the full problem can
be approximated by the full data of the trial objects. %of the discretization.
This allows very simple proofs of convergence rates for generalized finite
elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree
methods like Meshless Local Petrov-Galerkin techniques. It is also shown that
for a fixed trial space, weak formulations have a slightly better convergence
rate than strong formulations, but at the expense of numerical integration.
Since convergence rates are reduced to those coming from Approximation Theory,
and since trial spaces are arbitrary, this also covers various spectral and
pseudospectral methods. All of this is illustrated by examples.
