A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains

Elliott, Charles M.; Ranner, Thomas

doi:10.1093/imanum/draa062

Cited by 24 publications

(20 citation statements)

References 65 publications

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“…The assumption above is reasonable and satisfied by the usual choices of a Galerkin scheme; this can be seen with an easy calculation for the Fourier expansion, and in [26] for a finite element approximation. We now set up the Galerkin approximation for (CH s ) in these spaces L 2…”

Section: Galerkin Approximationmentioning

confidence: 97%

Cahn–Hilliard equations on an evolving surface

Caetano¹,

Elliott²

2021

Eur. J. Appl. Math

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We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

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Section: Galerkin Approximationmentioning

confidence: 97%

Cahn–Hilliard equations on an evolving surface

Caetano¹,

Elliott²

2021

Eur. J. Appl. Math

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“…In this section we wish to give an analysis of a time discrete system related to Problem 1.1. The discretisation of evolving space PDEs has been studied extensively in [16], the study of a fully discrete system for a heat equation was considered in [14] and for linear parabolic equations in [32] the full discretisation is also considered.…”

Section: Discretisation In Timementioning

confidence: 99%

“…The approach we develop allows the problem to be considered in its natural formulation. Handling the equations in natural spaces gives an elegance and simplicity which becomes particularly apparent in numerical implementation and finite element analysis [16] of PDEs on evolving surfaces/domains, while also being an interesting mathematical problem.…”

Section: Introductionmentioning

confidence: 99%

Non-stationary incompressible linear fluid equations in a moving domain

Djurdjevac¹,

Gräser²,

Herbert³

2021

Preprint

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This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a time discretisation of the system.

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“…[4,6,7,33,36,41,54,55]. A recent survey of surface FEM was published in [37], where both steady and moving surfaces are considered, the latter finding a unifying theory in [38].…”

Section: Introductionmentioning

confidence: 99%

Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

Bachini

Manzini

Putti

2021

Calcolo

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We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

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A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains

Cited by 24 publications

References 65 publications

Cahn–Hilliard equations on an evolving surface

Cahn–Hilliard equations on an evolving surface

Non-stationary incompressible linear fluid equations in a moving domain

Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

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