  The scalar, one-dimensional advection equation and heat equation are
considered. These equations are discretized in space, using a finite difference
method satisfying summation-by-parts (SBP) properties. To impose the boundary
conditions, we use a penalty method called simultaneous approximation term
(SAT). Together, this gives rise to two semi-discrete schemes where the
discretization matrices approximate the first and the second derivative
operators, respectively. The discretization matrices depend on free parameters
from the SAT treatment. We derive the inverses of the discretization matrices,
interpreting them as discrete Green's functions. In this direct way, we also
find out precisely which choices of SAT parameters that make the discretization
matrices singular. In the second derivative case, it is shown that if the
penalty parameters are chosen such that the semi-discrete scheme is dual
consistent, the discretization matrix can become singular even when the scheme
is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary
order of accuracy. For second and fourth order accurate operators, the inverses
are provided explicitly.
