Second order splitting for a class of fourth order equations

Elliott, Charles M.; Fritz, H. P.; Hobbs, Graham

doi:10.1090/mcom/3425

Cited by 4 publications

(2 citation statements)

References 14 publications

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“…We notice that it might be convenient to integrate by parts to remove the surface Hessian. This will give an alternate formula which is better suited for the numerical methods considered in [13,17]. Corollary 3.13.…”

Section: An Explicit Formula For the Derivativementioning

confidence: 99%

A formula for membrane mediated point particle interactions on near spherical biomembranes

Elliott

Herbert

2021

Interfaces Free Bound.

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Section: An Explicit Formula For the Derivativementioning

confidence: 99%

A formula for membrane mediated point particle interactions on near spherical biomembranes

Elliott

Herbert

2021

Interfaces Free Bound.

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“…There is a rich literature on coupling conditions typically prescribing contour and slope of the membrane either at particle boundaries [13,14,15] or in single points [16,17,18,19,20,21]. In a recently developed variational approach to hybrid models [22], see also [23,24], such coupling conditions take the role of constraints in energy minimization. While hybrid models are often formulated in the zerotemperature limit, effects of thermal fluctuations, mostly of the membrane, are about to attract more and more attention [25,26,27,28,29].…”

Section: Introductionmentioning

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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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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Second order splitting for a class of fourth order equations

Cited by 4 publications

References 14 publications

A formula for membrane mediated point particle interactions on near spherical biomembranes

A formula for membrane mediated point particle interactions on near spherical biomembranes

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

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

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