Second order splitting for a class of fourth order equations
Provides a theoretical framework for solving certain fourth-order surface equations, particularly relevant for biomembrane modeling, but the approach is presented in an abstract setting with limited concrete performance metrics.
The authors develop a well-posedness and approximation theory for a class of fourth-order elliptic PDEs by splitting them into coupled second-order saddle point problems, enabling treatment of non-smooth right-hand sides and operators with nontrivial kernels. Numerical experiments demonstrate the approach's applicability to biomembrane modeling and other domains.
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 surface equations 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.