  The symmetric $C^0$ interior penalty method is one of the most popular
discontinuous Galerkin methods for the biharmonic equation. This paper
introduces an automatic local selection of the involved stability parameter in
terms of the geometry of the underlying triangulation for arbitrary polynomial
degrees. The proposed choice ensures a stable discretization with guaranteed
discrete ellipticity constant. Numerical evidence for uniform and adaptive
mesh-refinement and various polynomial degrees supports the reliability and
efficiency of the local parameter selection and recommends this in practice.
The approach is documented in 2D for triangles, but the methodology behind can
be generalized to higher dimensions, to non-uniform polynomial degrees, and to
rectangular discretizations. Two appendices present the realization of our
proposed parameter selection in various established finite element software
packages as well as a detailed documentation of a self-contained MATLAB program
for the lowest-order $C^0$ interior penalty method.
