A Variational Approach to Particles in Lipid Membranes

Elliott, Charles M.; Gräser, Carsten; Hobbs, Graham; Kornhuber, Ralf; Wolf, Maren-Wanda

doi:10.1007/s00205-016-1016-9

Cited by 18 publications

(36 citation statements)

References 70 publications

(156 reference statements)

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“…Notice that we can treat Navier and periodic boundary conditions on ∂Ω using the same techniques (cf. [10]).…”

Section: Notation and Problem Formulationsmentioning

confidence: 99%

“…These hybrid models are based on a continuous surface description of the membrane while particles are described by discrete entities (see, e.g., [3,8,9,12,17]). For an overview on hybrid models we refer to [10] and the references cited therein. In the present paper we consider coupling conditions imposed on the particle boundaries and follow the notation introduced in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Gräser

Kies

2018

IMA Journal of Numerical Analysis

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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 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.

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“…Notice that we can treat Navier and periodic boundary conditions on ∂Ω using the same techniques (cf. [10]).…”

Section: Notation and Problem Formulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gräser

Kies

2018

IMA Journal of Numerical Analysis

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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: 2mentioning

confidence: 83%

“…Proof. This is shown in [14] by making use of the Lax-Milgram theorem with the coercivity of a over V .…”

Section: 2mentioning

confidence: 99%

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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“…Moreover the additional assumptions we make on b and m are quite natural for the applications we consider. See also [9,11] for other possible applications to fourth order partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Second order splitting for a class of fourth order equations

2019

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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 equations on closed surfaces 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.

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A Variational Approach to Particles in Lipid Membranes

Cited by 18 publications

References 70 publications

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

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

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

Second order splitting for a class of fourth order equations

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