Hamilton–Jacobi equations on an evolving surface

Deckelnick, Klaus; Elliott, Charles M.; Miura, Tatsu-Hiko; Styles, Vanessa

doi:10.1090/mcom/3420

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…Note that in (10) we can take the minimum since A x is compact. In general, classical solutions of ( 10)- (11) do not exist, and the unique viscosity solution is sought after [10]. A detailed discussion of viscosity solutions of Hamilton-Jacobi equations on manifolds can be found in [26].…”

Section: Static Hamilton-jacobi-bellman Equations On Surfacesmentioning

confidence: 99%

“…Theorem 2.4. If u : Γ → R is the viscosity solution to (10)- (11), then the constant normal extension of u, u : T ǫ → R, is the viscosity solution to (18)- (19).…”

Section: Extension Tomentioning

confidence: 99%

“…The set A x := S n−1 ∩ T x Γ is a compact set and we define A := S n−1 ∩ T M(Γ) to be the unit tangent bundle of Γ. Here r, f : Γ × A → R, T is a closed subset of Γ, and ∇ Γ u is the surface gradient on Γ [11].…”

Section: Introductionmentioning

confidence: 99%

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Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces

Martin¹,

Tsai²

2019

Preprint

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We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. In this work, we extend the optimal control problem given on the surface to an equivalent one defined in a sufficiently thin narrow band of the co-dimensional one surface. The extension is done appropriately so that the corresponding HJB equation, in the narrow band, has a unique viscosity solution which is identical to the constant normal extension of the value function of the original optimal control problem. With this framework, one can easily use existing (high order) numerical methods developed on Cartesian grids to solve HJB equations on surfaces, with a computational cost that scales with the dimension of the surfaces. This framework also provides a systematic way for solving HJB equations on the unstructured point clouds that are sampled from the surface.

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Section: Static Hamilton-jacobi-bellman Equations On Surfacesmentioning

confidence: 99%

“…Theorem 2.4. If u : Γ → R is the viscosity solution to (10)- (11), then the constant normal extension of u, u : T ǫ → R, is the viscosity solution to (18)- (19).…”

Section: Extension Tomentioning

confidence: 99%

See 1 more Smart Citation

Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces

Martin¹,

Tsai²

2019

Preprint

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“…[12,13,14,43,53,7,15] and the references cited therein). Other equations on moving surfaces were also studied such as a Stefan problem [4], a porous medium equation [5], the Cahn-Hilliard equation [17,9,42], and the Hamilton-Jacobi equation [10]. The authors of [56] formulated equations of nonlinear elasticity in an evolving ambient space like a moving surface.…”

mentioning

confidence: 99%

Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation

Miura¹

2022

Preprint

Self Cite

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We consider the Neumann type problem of the heat equation in a moving thin domain around a given closed moving hypersurface. The main result of this paper is an error estimate in the sup-norm for classical solutions to the thin domain problem and a limit equation on the moving hypersurface which appears in the thin-film limit of the heat equation. To prove the error estimate, we show a uniform a priori estimate for a classical solution to the thin domain problem based on the maximum principle. Moreover, we construct a suitable approximate solution to the thin domain problem from a classical solution to the limit equation based on an asymptotic expansion of the thin domain problem and apply the uniform a priori estimate to the difference of the approximate solution and a classical solution to the thin domain problem.

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“…Numerical approaches to solve these problems include surface finite elements, implicit surface formulations, diffuse interface approximations, trace finite elements, unfitted finite elements, finite volume schemes and mesh free methods. See the works of Dziuk (1988); Dziuk and Elliott (2007); Deckelnick, Dziuk, Elliott and Heine (2009); Dziuk and Elliott (2010); Deckelnick, Elliott and Ranner (2014); Deckelnick and Styles (2018); Olshanskii and Reusken (2017); Burman, Hansbo, Larson and Zahedi (2016); Lehrenfeld, Olshanskii and Xu (2018); Lehrenfeld and Olshanskii (2019); Giesselmann and Müller (2014); Deckelnick, Elliott, Miura and Styles (2019); Suchde and Kuhnert (2019) and the review of Dziuk and Elliott (2013a).…”

mentioning

confidence: 99%

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

Elliott

Ranner

2020

IMA Journal of Numerical Analysis

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We develop a unified theory for continuous-in-time finite element discretizations of partial differential equations posed in evolving domains, including the consideration of equations posed on evolving surfaces and bulk domains, as well as coupled surface bulk systems. We use an abstract variational setting with time-dependent function spaces and abstract time-dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described, which confirm the rates of convergence.

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Hamilton–Jacobi equations on an evolving surface

Cited by 3 publications

References 28 publications

Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces

Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces

Error estimate for classical solutions to the heat equation in a moving thin domain and its limit equation

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

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