scholarly journals Extension of the nonconforming Trefftz virtual element method to the Helmholtz problem with piecewise constant wave number

2020 ◽  
Vol 155 ◽  
pp. 160-180 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Alexander Pichler
Keyword(s):  
Wave Number ◽  
Virtual Element Method ◽  
Piecewise Constant ◽  
Helmholtz Problem ◽  
Virtual Element ◽  
Element Method

scholarly journals A nonconforming Trefftz virtual element method for the Helmholtz problem

2019 ◽  
Vol 29 (09) ◽  
pp. 1619-1656 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Convergence Rates ◽  
Plane Waves ◽  
Local Approximation ◽  
Approximation Properties ◽  
Virtual Element Method ◽  
Impedance Boundary ◽  
Helmholtz Problem ◽  
Text Version ◽  
Virtual Element ◽  
Element Method

We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e. functions belonging to the kernel of the Helmholtz operator. The global trial and test spaces are not fully discontinuous, but rather interelement continuity is imposed in a nonconforming fashion. Although their functions are only implicitly defined, as typical of the VEM framework, they contain discontinuous subspaces made of functions known in closed form and with good approximation properties (plane-waves, in our case). We carry out an abstract error analysis of the method, and derive [Formula: see text]-version error estimates. Moreover, we initiate its numerical investigation by presenting a first test, which demonstrates the theoretical convergence rates.

scholarly journals A plane wave virtual element method for the Helmholtz problem

2016 ◽  
Vol 50 (3) ◽  
pp. 783-808 ◽  
Author(s):  
Ilaria Perugia ◽  
Paola Pietra ◽  
Alessandro Russo
Keyword(s):  
Plane Wave ◽  
Virtual Element Method ◽  
Helmholtz Problem ◽  
Virtual Element ◽  
Element Method

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

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe 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.

Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions

2018 ◽  
Vol 77 (3) ◽  
pp. 1874-1908 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals The virtual element method for resistive magnetohydrodynamics

2021 ◽  
Vol 381 ◽  
pp. 113815
Author(s):  
S. Naranjo Alvarez ◽  
V. Bokil ◽  
V. Gyrya ◽  
G. Manzini
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

A Virtual Element Method for Crack Propagation

PAMM ◽  
2018 ◽  
Vol 18 (1) ◽  
Author(s):  
Ali Hussein ◽  
Blaž Hudobivnik ◽  
Fadi Aldakheel ◽  
Peter Wriggers ◽  
Pierre‐Alain Guidault ◽  
...  
Keyword(s):  
Crack Propagation ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method
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