  We modify the well-known interior penalty finite element discretization
method so that it allows for element-by-element assembly. This is possible due
to the introduction of additional unknowns associated with the interfaces
between neighboring elements. The resulting bilinear form, and a Schur
complement (reduced) version of it, are utilized in a number of auxiliary space
preconditioners for the original conforming finite element discretization
problem. These preconditioners are analyzed on the fine scale and their
performance is illustrated on model second order scalar elliptic problems
discretized with high order elements.
