# Discretization error estimates for penalty formulations of a linearized Canham-Helfrich type energy

## TL;DR

This paper develops error estimates for penalty-based finite element methods solving a linearized Canham-Helfrich energy problem, crucial for modeling biomembranes with embedded particles, without needing to resolve particle boundaries.

## Contribution

It provides the first discretization error estimates for penalty formulations of a fourth-order membrane energy problem with reduced regularity solutions.

## Key findings

- Proves almost-H^{5/2} regularity of solutions.
- Derives optimal error estimates for penalty finite element discretizations.
- Numerical results confirm theoretical error bounds.

## Abstract

This paper is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of biomembranes with embedded particles. There, the curve conditions result from the imposed particle--membrane coupling. We prove almost-$H^{\frac{5}{2}}$ regularity of the solution and then consider two possible penalty formulations. For the combination of these penalty formulations with a Bogner-Fox-Schmit finite element discretization we prove discretization error estimates which are optimal in view of the solution's reduced regularity. The error estimates are based on a general estimate for linear penalty problems in Hilbert spaces. Finally, we illustrate the theoretical results by numerical computations. An important feature of the presented discretization is that it does not require to resolve the particle boundary. This is crucial in order to avoid re-meshing if the presented problem arises as subproblem in a model where particles are allowed to move or rotate.