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

TL;DR

This paper develops a novel splitting approach for solving a class of fourth order PDEs with point constraints, using second order equations and penalty methods, with applications in biomembrane modeling.

Contribution

It introduces a well-posedness and approximation framework for constrained fourth order PDEs via splitting into coupled second order problems.

Findings

The method effectively handles point constraints in fourth order PDEs.

Numerical experiments demonstrate the approach's accuracy and applicability.

The framework is applicable to biomembrane modeling and other surface PDEs.

Abstract

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.