Analytic Regularity for the Incompressible Navier--Stokes Equations in Polygons

Marcati, Carlo; Schwab, Christoph

doi:10.1137/19m1247334

Cited by 13 publications

(11 citation statements)

References 26 publications

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“…The following result is a finite regularity shift result in weighted Sobolev spaces for solutions to the Stokes problem; see [27,Theorem 5.7] and [32,Section 5]; see also [35,Proposition 1.8] for the case of homogeneous spaces.…”

Section: Regularity Of the Solution To (1)mentioning

confidence: 99%

“…This is not the case when the domain of the equation is polygonal/polyhedral. In fact, even with smooth data, solutions are expected to have singularities at the corners of the domain; see, e.g., [27,35]. More precisely, it can be proven that they belong to Kondrat'ev spaces, i.e., weighted Sobolev spaces with weight given by a function of the distance from the corners of the domain; see definitions (4) and (5) below.…”

Section: Introductionmentioning

confidence: 99%

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p- and hp- virtual elements for the Stokes problem

2021

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We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

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Section: Regularity Of the Solution To (1)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

2021

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show abstract

“…The following result is a finite regularity shift result in weighted Sobolev spaces for solutions to the Stokes problem; see [26,Theorem 5.7] and [31, Section 5]; see also [35,Proposition 1.8] for the case of homogeneous spaces.…”

Section: Regularity Of the Solutionmentioning

confidence: 99%

“…This is not the case when the domain of the equation is polygonal/polyhedral. In fact, even with smooth data, solutions are expected to have singularities at the corners of the domain; see, e.g., [26,35]. More precisely, it can be proven that they belong to Kondrat'ev spaces, i.e., weighted Sobolev spaces with weight given by a function of the distance from the corners of the domain; see definitions (4) and ( 5) below.…”

Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

Chernov¹,

Marcati²,

Mascotto³

2020

Preprint

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We analyse the p-and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poissonlike VE spaces. The upside of this fact is that we inherit from [7] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p-and hp-versions of the method.

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“…This estimate is proven in Theorem 2 and constitutes a result of independent interest. For previous weighted analytic regularity results for elliptic problems, we refer, among others, to [GS06,CDN12] for the linear case, and to [MS19] for the analysis -based on the same arguments as in the present paper -of the (nonlinear) two dimensional Navier-Stokes equations.…”

mentioning

confidence: 99%

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

Maday¹,

Marcati²

2019

Preprint

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A. We study a class of nonlinear eigenvalue problems of Scrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined hp discontinuous Galerkin (dG) method.We show that, for weighted analytic potentials and for up-to-quartic nonlinearities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined hp space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. In addition, we investigate the role of pointwise convergence in the doubling of the convergence rate for the eigenvalues with respect to the convergence rate of eigenfunctions. We conclude with a series of numerical tests to validate the theoretical results.

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Analytic Regularity for the Incompressible Navier--Stokes Equations in Polygons

Cited by 13 publications

References 26 publications

p- and hp- virtual elements for the Stokes problem

p- and hp- virtual elements for the Stokes problem

p- and hp- virtual elements for the Stokes problem

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

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