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We formulate a well-posedness and approximation theory for a class ofgeneralised saddle point problems with a specific form of constraints. In thisway we develop an approach to a class of fourth order elliptic partialdifferential equations with point constraints using the idea of splitting intocoupled second order equations. An approach is formulated using a penaltymethod to impose the constraints. Our main motivation is to treat certainfourth order equations involving the biharmonic operator and point Dirichletconstraints for example arising in the modelling of biomembranes on curved andflat surfaces but the approach may be applied more generally. The theory forwell-posedness and approximation is presented in an abstract setting. Severalexamples are described together with some numerical experiments.

Second order splitting of a class of fourth order PDEs with point constraints