  All finite element methods, as well as much of the Hilbert-space theory for
partial differential equations, rely on variational formulations, that is,
problems of the type: find $u\in V$ such that $a(v,u) = l(v)$ for each $v\in
L$, where $V, L$ are Sobolev spaces. However, for systems of Friedrichs type,
there is a sharp disparity between established well-posedness theories, which
are not variational, and the very successful discontinuous Galerkin methods
that have been developed for such systems, which are variational. In an attempt
to override this dichotomy, we present, through three specific examples of
increasing complexity, well-posed variational formulations of boundary and
initial--boundary-value problems of Friedrichs type. The variational forms we
introduce are generalizations of those used for discontinuous Galerkin methods,
in the sense that inhomogeneous boundary and initial conditions are enforced
weakly through integrals in the variational forms. In the variational forms we
introduce, the solution space is defined as a subspace $V$ of the graph space
associated with the differential operator in question, whereas the test
function space $L$ is a tuple of $L^2$ spaces that separately enforce the
equation, boundary conditions of characteristic type, and initial conditions.
