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

Gräser, Carsten; Kies, Tobias

doi:10.1093/imanum/drx071

Cited by 6 publications

(7 citation statements)

References 12 publications

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“…For f = 0, these are the membrane problem studied in [14,18]. In very much the same way as the preceding subsection, one may see that the point constraint problem can be written as the following PDE in distribution…”

Section: Fourth Order Formulationmentioning

confidence: 83%

“…For this flat problem, we consider the Monge-Gauge energy [14]. The numerical analysis for this has been considered in [18] for finite size particles with constraints on closed curves using a penalty method. The authors make use of higher order H 2 conforming finite elements so do not need to split the equation.…”

Section: 2mentioning

confidence: 99%

“…Our abstract formulation is motivated by applications of this theory to fourth order boundary value problems arising in the modelling of biomembranes posed on a flat domain, sphere or torus, with a specific example focusing on the sphere for ease of exposition. The problems are derived in [12,14,15,18] as approximations of minimisers of the Helfrich energy [19] with point constraints. In this context these arise as Dirichlet constraints on the membrane deformation modelling the attachment of point particles to the membrane at fixed locations.…”

Section: Introductionmentioning

confidence: 99%

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

Elliott

Herbert

2020

Math. Comp.

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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.In particular we have in mind an example of the formwhere c 0 , c 1 are bounded but c(·, ·) is not coercive and a linear map T : W 1,q (Γ) → R N defined by (T η) j := η(X j ), j = 1, ..., N with X j ∈ Γ, j = 1, ..., N .1.1. Background. The study of saddle point problems is well documented, [4,16], with many applications, for example in fluid mechanics, [17], or in linear elasticity, [2]. Note that in many of the cases in which m = 0, the authors require some strong assumptions on c, at least positive semi definite, see [4,9,21]. The system (1.1) is an extension to that considered in [13]. The extended system is posed over an affine subspace of X × Y , rather than over the whole space. If in (1.1), we were to seach for a solution in X 0 × Y , the first equation were to be considered with test functions in X 0 and the third equation to be dropped, this recovers the abstract system studied in [13]. We will use the assumptions made in [13] together with an additional assumption to handle the constraint. The assumptions will be given in Section 2.In [7], the authors consider the approximation of Stokes flow by penalising the incompressibility condition. In particular, they show that the penalty terms approximate the pressure. We also consider an abstract problem with penalty and show that, in our setting, the penalty terms converge to the Lagrange multiplier associated to the constraints. Further to this, we show estimates between the solution to the problem with penalised constraint and the solution to the problem with enforced constraint. An abstract finite element theory with error bounds is presented. The results of this paper extend those of of [13] where for example, in [13, Section 6 and 7] it is shown that the well posedness theory in that paper may be applied to a problem with penalised point constraints without consideration of the convergence with respect to the penalty parameter.The motivation for the abstract setting is to handle second order splitting for a class of fourth order surface PDEs with point Dirichlet constraints arising in the modelling of biomembranes, [12]. The setting of [13] may be directly applied to the penalty approximation for fixed penalty parameter but does not handle the hard...

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Section: Fourth Order Formulationmentioning

confidence: 83%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Elliott

Herbert

2020

Math. Comp.

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“…For the construction and analysis of finite element approximations of the minimization problem (7), and thus of M(q), we refer, e.g., to [22,37,42,43].…”

Section: Finite-size Particles With Curve Constraintsmentioning

confidence: 99%

“…As a consequence, discretization error estimates for suitable finite element approximations of u q directly carry over to ∂ e M(q). We refer to [42,43] for details. Such kind of properties are not available for straightforward finite difference approximations.…”

Section: Differentiability and Stable Representation Of Gradientmentioning

confidence: 99%

Free energy computation of particles with membrane-mediated interactions via Langevin dynamics

Kies¹,

Gräser²,

Site³

et al. 2020

Preprint

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We apply well-established concepts of Langevin sampling to derive a new class of algorithms for the efficient computation of free energy differences of fluctuating particles embedded in a 'fast' membrane, i.e., a membrane that instantaneously adapts to varying particle positions. A geometric potential accounting for membrane-mediated particle interaction is derived in the framework of variational hybrid models for particles in membranes. Recent explicit representations of the gradient of the geometric interaction potential allows to apply well-known gradient based Markov Chain Monte-Carlo (MCDC) methods such as Langevin-based sampling.

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