A C1 virtual element method for the stationary quasi-geostrophic equations of the ocean

Mora, David; Silgado, Alberth

doi:10.1016/j.camwa.2021.05.022

Cited by 14 publications

(6 citation statements)

References 37 publications

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“…The resulting conforming formulation is shown to be well-posed through a Banach fixed-point argument provided that the mesh size is small enough, and optimal errors bounds are proved when the error is measured in the H 2 norm. Interesting applications of highly-regular conforming VEM can be also found in the context of geostrophic equations 61 and fourth-order subdiffusion equations. 54 Highly-regular conforming VEMs have been recently proposed and analyzed for general polyharmonic boundary value problems 10 of the form (−∆) p1 u = f , p 1 ≥ 1, in two dimensions.…”

Section: Background Materials On Arbitrarily Regular Virtual Element ...mentioning

confidence: 99%

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

Antonietti

Manzini

Scacchi

et al. 2021

Math. Models Methods Appl. Sci.

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The virtual element method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order [Formula: see text], for any integer [Formula: see text]. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of [Formula: see text], [Formula: see text] being the computational domain and [Formula: see text] another suitable integer number. In this review, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order regularity on [Formula: see text]. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases [Formula: see text] and [Formula: see text], respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

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Section: Background Materials On Arbitrarily Regular Virtual Element ...mentioning

confidence: 99%

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

Antonietti

Manzini

Scacchi

et al. 2021

Math. Models Methods Appl. Sci.

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show abstract

“…The VEM introduced in [12] as a generalization of FEM which is characterized by the capability of dealing with very general polygonal/polyhedral meshes, including hanging nodes and nonconvex elements (see [10,7,13,15,19,20,26,27,30,32,33,37] and refereneces therein). The VEM also permits to easily implement highly regular conforming discrete spaces [18,22] which make the method very feasible to solve various fourth-order problems [8,35,14,34,36]. Regarding VEM for time dependent problems, we mention the following works [2,1,4,6,9,39,38,40].…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

Adak¹,

Mora²,

Natarajan³

2022

Preprint

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In this article, we have considered a nonlinear nonlocal time dependent fourth order equation demonstrating the deformation of a thin and narrow rectangular plate. We propose C 1 conforming virtual element method (VEM) of arbitrary order, k ≥ 2, to approximate the model problem numerically. We employ VEM to discretize the space variable and fully implicit scheme for temporal variable. Well-posedness of the fully discrete scheme is proved under certain conditions on the physical parameters, and we derive optimal order of convergence in both space and time variable. Finally, numerical experiments are presented to illustrate the behaviour of the proposed numerical scheme.

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“…A VEM for Helmholtz problems based on non-conforming approximation spaces of Trefftz functions, i.e., functions that belong to the kernel of the Helmholtz operator, is found [51] The first works using a 𝐶 1 -regular conforming VEM addressed the classical plate bending problems [33,38], second-order elliptic problems [24,25], and the nonlinear Cahn-Hilliard equation [3]. More recently, highly regular virtual element spaces were considered for the von Kármán equation modelling the deformation of very thin plates [49], geostrophic equations [53] and fourth-order subdiffusion equations [48], two-dimensional plate vibration problem of Kirchhoff plates [52], the transmission eigenvalue problems [54] the fourth-order plate buckling eigenvalue problem [55]. In [10], we proposed the highly-regular conforming VEM for the two-dimensional polyharmonic problem (−Δ) 𝑝 1 𝑢 = 𝑓 , 𝑝 1 ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

Antonietti¹,

Manzini²,

Scacchi³

et al. 2021

Preprint

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The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order 𝑝 1 using conforming, finite dimensional subspaces of 𝐻 𝑝 2 (Ω), where 𝑝 1 and 𝑝 2 are two integer numbers such that 𝑝 2 ≥ 𝑝 1 ≥ 1 and Ω ∈ R 2 is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects such as the choice and unisolvence of the degrees of freedom are discussed, also providing specific examples corresponding to various practical cases of high global regularity. Finally, the construction of the "enhanced" formulation of the virtual element spaces is also discussed in details with a proof that the dimension of the "regular" and "enhanced" spaces is the same and that the virtual element functions in both spaces can be described by the same choice of the degrees of freedom.

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A C1 virtual element method for the stationary quasi-geostrophic equations of the ocean

Cited by 14 publications

References 37 publications

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

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