  We consider finite difference approximations of the second derivative,
exemplified in Poisson's equation, the heat equation and the wave equation. The
finite difference operators satisfy a summation-by-parts property, which mimics
the integration-by-parts. Since the operators approximate the second
derivative, they are singular by construction. To impose boundary conditions,
these operators are modified using Simultaneous Approximation Terms. This makes
the modified matrices non-singular, for most choices of boundary conditions.
Recently, inverses of such matrices were derived. However, when considering
Neumann boundary conditions on both boundaries, the modified matrix is still
singular. For such matrices, we have derived an explicit expression for the
Moore-Penrose pseudoinverse, which can be used for solving elliptic problems
and some time-dependent problems. The condition for this new pseudoinverse to
be valid, is that the modified matrix does not have more than one zero
eigenvalue. We have reconstructed the sixth order accurate narrow-stencil
operator with a free parameter and show that more than one zero eigenvalue can
occur. We have performed a detailed analysis on the free parameter to improve
the properties of the second derivative operator. We complement the derivations
by numerical experiments to demonstrate the improvements of the new second
derivative operator.
