Second order splitting for a class of fourth order equations

TL;DR

This paper introduces a novel splitting method for solving a class of fourth order elliptic PDEs by decomposing them into coupled second order equations, with applications to biomembrane modeling.

Contribution

It develops a well-posedness and approximation framework for generalized saddle point problems, enabling effective treatment of complex fourth order equations.

Findings

The approach is well-posed and convergent.

Numerical experiments demonstrate effectiveness.

Applicable to equations with non-smooth data.

Abstract

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.