A fully discrete virtual element scheme for the Cahn–Hilliard equation in mixed form

Liu, Xin; He, Zhengkang; Chen, Zhangxin

doi:10.1016/j.cpc.2019.106870

Cited by 17 publications

(8 citation statements)

References 66 publications

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“…As considered in e.g. [3,13,34], for this experiment we monitor the evolution of initial data relating to a cross-shaped interface between phases. The initial data is described as follows.…”

Section: Test 1: General Convergencementioning

confidence: 99%

“…More recently, we have seen a handful of virtual element methods to discretize the Cahn-Hilliard equation [3,33,34]. The virtual element method is an extension and generalization of both finite element and mimetic difference methods.…”

Section: Introductionmentioning

confidence: 99%

“…Both works present a continuous-in-time (semidiscrete) convergence result for the lowest order VEM space. The only other application of the virtual element method to the Cahn-Hilliard equation is seen in [34] where a fully discrete scheme is presented and is shown to satisfy both a discrete energy law and a mass conservation law. In contrast to the virtual element discretizations developed in [3,33,34], we present the first analysis of a higher order VEM method and achieve optimal order error estimates for the semidiscrete scheme.…”

Section: Introductionmentioning

confidence: 99%

“…The only other application of the virtual element method to the Cahn-Hilliard equation is seen in [34] where a fully discrete scheme is presented and is shown to satisfy both a discrete energy law and a mass conservation law. In contrast to the virtual element discretizations developed in [3,33,34], we present the first analysis of a higher order VEM method and achieve optimal order error estimates for the semidiscrete scheme. A clear advantage of using higher order methods is the ability to use considerably lower grid resolutions which in turn can avoid the use of strongly adapted grids.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

Dedner¹,

Hodson²

2021

Preprint

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In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the continuous-in-time scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

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Section: Test 1: General Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

Dedner¹,

Hodson²

2021

Preprint

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“…So far the Virtual Element Method has been successfully applied to many important physical applications. In particular, recent efforts have been devoted to show the accuracy and advantages of this method in the numerical approximation of nonlinear problems such as the Cahn-Hilliard equation [8,21], models in cardiology [7], bulk-surface reaction-diffusion systems [19], and semilinear elliptic [10,5], hyperbolic [2] and parabolic [1] equations.…”

Section: Introductionmentioning

confidence: 99%

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

Gómez¹

2021

Preprint

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We propose an efficient method for the numerical solution of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method (VEM), which not only significantly reduces the number of degrees of freedom compared to the classical VEM but also, under certain conditions on the mesh allows to approximate the nonlinear term with an interpolant in the Serendipity VEM space; which substantially improves the efficiency of the method. An error analysis for the semi discrete formulation is carried out, and an optimal estimate for the error in the L 2 -norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with high order approximations up to order 6.

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A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

Meng

Wang

Mei

2021

Calcolo

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A fully discrete virtual element scheme for the Cahn–Hilliard equation in mixed form

Cited by 17 publications

References 66 publications

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

A higher order nonconforming virtual element method for the Cahn-Hilliard equation

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

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