  We formulate a well-posedness and approximation theory for a class of
generalised saddle point problems with a specific form of constraints. In this
way we develop an approach to a class of fourth order elliptic partial
differential equations with point constraints using the idea of splitting into
coupled second order equations. An approach is formulated using a penalty
method to impose the constraints. Our main motivation is to treat certain
fourth order equations involving the biharmonic operator and point Dirichlet
constraints for example arising in the modelling of biomembranes on curved and
flat surfaces but the approach may be applied more generally. The theory for
well-posedness and approximation is presented in an abstract setting. Several
examples are described together with some numerical experiments.
