Second order splitting for a class of fourth order equations

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order equations on closed surfaces arising in the modelling of biomembranes but the approach may be applied more generally. In particular we are interested in equations with non-smooth right-hand sides and operators which have non-trivial kernels. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.