  We present a framework for solving the triharmonic equation over bilinearly
parameterized planar multi-patch domains by means of isogeometric analysis. Our
approach is based on the construction of a globally $C^2$-smooth isogeometric
spline space which is used as discretization space. The generated $C^2$-smooth
space consists of three different types of isogeometric functions called patch,
edge and vertex functions. All functions are entirely local with a small
support, and numerical examples indicate that they are well-conditioned. The
construction of the functions is simple and works uniformly for all multi-patch
configurations. While the patch and edge functions are given by a closed form
representation, the vertex functions are obtained by computing the null space
of a small system of linear equations. Several examples demonstrate the
potential of our approach for solving the triharmonic equation.
