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.
Mots-clés : Surface PDEs, Laplace-Beltrami equation, surface finite element method, Virtual Element Method
@article{M2AN_2018__52_3_965_0,
author = {Frittelli, Massimo and Sgura, Ivonne},
title = {Virtual {Element} {Method} for the {Laplace-Beltrami} equation on surfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {965--993},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2017040},
mrnumber = {3865555},
zbl = {1456.65160},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017040/}
}
TY - JOUR AU - Frittelli, Massimo AU - Sgura, Ivonne TI - Virtual Element Method for the Laplace-Beltrami equation on surfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 965 EP - 993 VL - 52 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017040/ DO - 10.1051/m2an/2017040 LA - en ID - M2AN_2018__52_3_965_0 ER -
%0 Journal Article %A Frittelli, Massimo %A Sgura, Ivonne %T Virtual Element Method for the Laplace-Beltrami equation on surfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 965-993 %V 52 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017040/ %R 10.1051/m2an/2017040 %G en %F M2AN_2018__52_3_965_0
Frittelli, Massimo; Sgura, Ivonne. Virtual Element Method for the Laplace-Beltrami equation on surfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 965-993. doi : 10.1051/m2an/2017040. http://archive.numdam.org/articles/10.1051/m2an/2017040/
[1] and , Basic principles of virtual element methods. Math. Mod. Methods Appl. Sci. 23 (2013) 199–214 | DOI | MR | Zbl
[2] and , Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812 | DOI | MR | Zbl
[3] , and , A virtual element method for the Steklov eigenvalue problem. Math. Mod. Methods Appl. Sci. 25 (2015) 1421–1445 | DOI | MR | Zbl
[4] and Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Diff. Eq. 31 (2015) 2110–2134 | DOI | MR | Zbl
[5] , Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74 (2017) 882–898 | DOI | MR | Zbl
[6] , and , A C1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54 (2016) 34–56 | DOI | MR | Zbl
[7] , and , A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elements Anal. Design 109 (2016) 23–36 | DOI
[8] and , The hitchhiker’s guide to the virtual element method. Math. Mod. Methods Appl. Sci. 24 (2014) 1541–1573 | DOI | MR | Zbl
[9] and , nonconforming virtual element method. ESAIM: M2AN 50 (2016) 879–904 | DOI | Numdam | MR | Zbl
[10] , and , Anisotropic constitutive model of plasticity capable of accounting for details of meso-structure of two-phase composite material. Comput. Struct. 90 (2012) 153–162 | DOI
[11] , A memory efficient discontinuous Galerkin finite-element time-domain scheme for simulations of finite periodic structures. Microwave Optical Technol. Lett. 56 (2014) 1929–1933 | DOI
[12] , , and , Interactive mesh fusion based on local 3d metamorphosis. Graphics Interface 99 (1999) 148–156
[13] , , and , Snappaste: an interactive technique for easy mesh composition. Visual Comput. 22 (2006) 835–844 | DOI
[14] , and , Adaptive discontinuous Galerkin methods for nonstationary convection–diffusion problems. IMA J. Numer. Anal. 34 (2014) 1578–1597 | DOI | MR | Zbl
[15] , and , An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements Anal. Design 43 (2007) 847–860 | DOI | MR
[16] and , A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781 | DOI | MR | Zbl
[17] and , Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253 (2013) 455–462 | DOI | MR | Zbl
[18] and , Transport schemes on a sphere using radial basis functions. J. Comput. Phys. 226 (2007) 1059–1084 | DOI | MR | Zbl
[19] and , A radial basis function method for the shallow water equations on a sphere, In Proc. of the Royal Society of London A: Math., Physical and Engineering Sciences Proc. R. Soc. A 465 (2009) 1949–1976 | MR | Zbl
[20] , , and , Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method. Phys. Rev. E 72 (2005) 016710 | DOI
[21] , , and , Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174 (2001) 759–780 | DOI | MR | Zbl
[22] , and , A lagrangian particle method for reaction–diffusion systems on deforming surfaces. J. Math. Biology 61 (2010) 649–663 | DOI | MR | Zbl
[23] , and , The surface finite element method for pattern formation on evolving biological surfaces. J. Math. Biology 63 (2011) 1095–1119 | DOI | MR | Zbl
[24] and , A high-order kernel method for diffusion and reaction-diffusion equations on surfaces. J. Scientific Comput. 56 (2013) 535–565 | DOI | MR | Zbl
[25] , , and , Preserving invariance properties of reaction-diffusion systems on stationary surfaces. To appear in: IMA J. Num. Anal. (2017), drx058 | MR | Zbl
[26] , and , Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J. Math. Biology 42 (2001) 387–423 | DOI | MR | Zbl
[27] and , Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys. 229 (2010) 6585–6612 | DOI | MR | Zbl
[28] , and , Modelling cell motility and chemotaxis with evolving surface finite elements. J. Royal Soc. Interface, 9 (2012) 3027–3044 | DOI
[29] and , Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations. Math. Comput. 74 1257–1280 (2005) | DOI | MR | Zbl
[30] and , Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys. 227 (2008) 9727–9741 | DOI | MR | Zbl
[31] , and , Discrete surface modelling using partial differential equations. Computer Aided Geometric Design 23 (2006) 125–145 | DOI | MR | Zbl
[32] Partial differential equations III: Nonlinear Equations, 2ndEd. Vol 117 of Applied Math. Sciences, Springer (2011) | MR | Zbl
[33] and , The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput. 31 (2009) 4330–4350 | DOI | MR | Zbl
[34] and , A finite volume method on general surfaces and its error estimates. J. Math. Analy. Appl. 352 (2009) 645–668 | DOI | MR | Zbl
[35] , and , Anal. of the discontinuous Galerkin method for elliptic problems on surfaces. IMA J. Numer. Anal., (2013) drs033.. | MR | Zbl
[36] and , Geometric error of finite volume schemes for conservation laws on evolving surfaces. Numer. Math. 128 (2014) 489–516 | DOI | MR | Zbl
[37] and , Finite element methods for surface PDEs. Acta Numer. 22 (2013) 289–396 | DOI | MR | Zbl
[38] , and , Projected finite elements for reaction–diffusion systems on stationary closed surfaces. Appl. Numer. Math. 96 (2015) 45–71 | DOI | MR | Zbl
[39] , Finite elements for the Beltrami operator on arbitrary surfaces. Partial Diff. Equ. Calcul. Variat. (1988) 142–155 | DOI | MR | Zbl
[40] Partial differential equations I: Basic Theory, 2ndEd., n Vol. 115 of Series: Appl. Math. Sci. Springer (2011) | MR | Zbl
[41] and Numerical approximation of partial differential equations. In Vol. 23 of Springer Science & Business Media (2008) | Zbl
[42] , , , and , Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391 | DOI | MR | Zbl
[43] and The mathematical theory of finite element methods. In Vol. 15 of , Springer Science & Business Media (2007) | MR | Zbl
[44] The finite element method for elliptic problems. SIAM (2002) | DOI | MR
[45] , Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47 (2009) 805–827 | DOI | MR | Zbl
[46] , On the symmetries of spherical harmonics. Can. J. Math 6 (1954) 135–157 | DOI | MR | Zbl
[47] Mesh generation for implicit geometries. Ph.D. Thesis, Massachusetts Institute of Technology (2004) | MR
[48] Advanced techniques for the generation and the gdaptation of complex surface meshes. Ph.D. Thesis, Politecnico di Milano (2014)
[49] and , A simple mesh generator in MATLAB. SIAM Rev. 46 (2004) 329–345 | DOI | MR | Zbl
[50] ALBERTA - An adaptive hierarchical finite element toolbox, http://www.alberta-fem.de.
[51] and , Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175–202 | DOI | MR | Zbl
[52] and , An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421–442 | DOI | MR | Zbl
Cité par Sources :
