Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems
The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 399-416.

We consider, as a simple model problem, the application of Virtual Element Methods (VEM) to the linear Magnetostatic three-dimensional problem in the classical Vector Potential formulation. The Vector Potential is treated as a triplet of 0-forms, approximated by nodal VEM spaces. However this is not done using three classical H 1 -conforming nodal Virtual Elements, and instead we use the Stokes Elements introduced originally in the paper Divergence free Virtual Elements for the Stokes problem on polygonal meshes (ESAIM Math. Model. Numer. Anal. 51 (2017), 509–535) for the treatment of incompressible fluids.

Published online:
DOI: 10.5802/smai-jcm.40
Classification: 65N30
Keywords: Virtual Element Methods, Serendipity, Magnetostatic problems, Vector Potential
Lourenço Beirão da Veiga 1; Franco Brezzi 2; L. Donatella Marini 3; Alessandro Russo 1

1 Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, Via Cozzi 55, I-20153, Milano, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy
2 IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy
3 Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100 Pavia, Italy and IMATI-CNR, Via Ferrata 1, I-27100 Pavia, Italy
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{SMAI-JCM_2018__4__399_0,
     author = {Louren\c{c}o Beir\~ao da Veiga and Franco Brezzi and L. Donatella Marini and Alessandro Russo},
     title = {Virtual {Element} approximations of the {Vector} {Potential} {Formulation} of {Magnetostatic} problems},
     journal = {The SMAI Journal of computational mathematics},
     pages = {399--416},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {4},
     year = {2018},
     doi = {10.5802/smai-jcm.40},
     zbl = {1416.78024},
     mrnumber = {3883675},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.40/}
}
TY  - JOUR
AU  - Lourenço Beirão da Veiga
AU  - Franco Brezzi
AU  - L. Donatella Marini
AU  - Alessandro Russo
TI  - Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems
JO  - The SMAI Journal of computational mathematics
PY  - 2018
SP  - 399
EP  - 416
VL  - 4
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.40/
DO  - 10.5802/smai-jcm.40
LA  - en
ID  - SMAI-JCM_2018__4__399_0
ER  - 
%0 Journal Article
%A Lourenço Beirão da Veiga
%A Franco Brezzi
%A L. Donatella Marini
%A Alessandro Russo
%T Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems
%J The SMAI Journal of computational mathematics
%D 2018
%P 399-416
%V 4
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.40/
%R 10.5802/smai-jcm.40
%G en
%F SMAI-JCM_2018__4__399_0
Lourenço Beirão da Veiga; Franco Brezzi; L. Donatella Marini; Alessandro Russo. Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems. The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 399-416. doi : 10.5802/smai-jcm.40. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.40/

[1] P. F. Antonietti; L. Beirão da Veiga; S. Scacchi; M. Verani A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., Volume 54 (2016) no. 1, pp. 34-57 | DOI | MR | Zbl

[2] P. F. Antonietti; M. Bruggi; S. Scacchi; M. Verani On the virtual element method for topology optimization on polygonal meshes: A numerical study, Comput. Math. Appl., Volume 74 (2017) no. 5, pp. 1091 -1109 | DOI | MR | Zbl

[3] D. N. Arnold; G. Awanou Finite element differential forms on cubical meshes, Math. Comp., Volume 83 (2014) no. 288, pp. 1551-1570 | DOI | MR | Zbl

[4] D. N. Arnold; R. S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155 | DOI | MR | Zbl

[5] M. Arroyo; M. Ortiz Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, Internat. J. Numer. Methods Engrg., Volume 65 (2006) no. 13, pp. 2167-2202 | DOI | MR | Zbl

[6] E. Artioli; S. de Miranda; C. Lovadina; L. Patruno A stress/displacement virtual element method for plane elasticity problems, Comput. Methods Appl. Mech. Engrg., Volume 325 (2017), pp. 155-174 | DOI | MR | Zbl

[7] F. Assous; P. Ciarlet; E. Sonnendrücker Resolution of the Maxwell equations in a domain with reentrant corners, RAIRO Modél. Math. Anal. Numér., Volume 32 (1998) no. 3, pp. 359-389 | DOI | Numdam | MR | Zbl

[8] B. Ayuso; K. Lipnikov; G. Manzini The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 879-904 | DOI | MR | Zbl

[9] I. Babuška; U. Banerjee; J. E. Osborn Survey of meshless and generalized finite element methods: a unified approach, Acta Numer., Volume 12 (2003), pp. 1-125 | DOI | MR

[10] S. Badia; R. Codina A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions, SIAM J. Numer. Anal., Volume 50 (2012) no. 2, pp. 398-417 | DOI | MR | Zbl

[11] L. Beirão da Veiga; F. Brezzi; A. Cangiani; G. Manzini; L. D. Marini; A. Russo Basic principles of virtual element methods, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 1, pp. 199-214 | DOI | MR | Zbl

[12] L. Beirão da Veiga; F. Brezzi; F. Dassi; L. D. Marini; A. Russo Virtual Element approximation of 2D magnetostatic problems, Comput. Methods Appl. Mech. Engrg., Volume 327 (2017), pp. 173-195 | DOI | MR | Zbl

[13] L. Beirão da Veiga; F. Brezzi; F. Dassi; L. D. Marini; A. Russo Lowest order Virtual Element approximation of magnetostatic problems, Comput. Methods Appl. Mech. Engrg., Volume 332 (2018), pp. 343-362 | DOI | MR | Zbl

[14] L. Beirão da Veiga; F. Brezzi; F. Dassi; L. D. Marini; A. Russo Serendipity Virtual Elements for General Elliptic Equations in Three Dimensions, Chinese Annals of Mathematics Series B, Volume 39 (2018) no. 2, pp. 315-334 | DOI | MR | Zbl

[15] L. Beirão da Veiga; F. Brezzi; L. D. Marini; A. Russo The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 8, pp. 1541-1573 | DOI | MR | Zbl

[16] L. Beirão da Veiga; F. Brezzi; L. D. Marini; A. Russo H(div) and H(curl)-conforming VEM, Numer. Math., Volume 133 (2016) no. 2, pp. 303-332 | Zbl

[17] L. Beirão da Veiga; F. Brezzi; L. D. Marini; A. Russo Serendipity Nodal VEM spaces, Comp. Fluids, Volume 141 (2016), pp. 2-12 | DOI | MR | Zbl

[18] L. Beirão da Veiga; F. Brezzi; L. D. Marini; A. Russo Serendipity Face and Edge VEM spaces, Rend. Lincei Mat. Appl., Volume 28 (2017) no. 1, pp. 143-180 | MR | Zbl

[19] L. Beirão da Veiga; C. Lovadina; A. Russo Stability Analysis for the Virtual Element Method, Math. Models Methods Appl. Sci., Volume 27 (2017) no. 13, pp. 2557-2594 | DOI | MR | Zbl

[20] L. Beirão da Veiga; C. Lovadina; G. Vacca Divergence free Virtual Elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal., Volume 51 (2017), pp. 509-535 | DOI | MR

[21] L. Beirão da Veiga; C. Lovadina; G. Vacca Virtual Elements for the Navier-Stokes problem on polygonal meshes, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1210-1242 | DOI | MR | Zbl

[22] M. F. Benedetto; S. Berrone; S. Pieraccini; S. Scialò The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Engrg., Volume 280 (2014), pp. 135-156 | DOI | MR | Zbl

[23] M. F. Benedetto; S. Berrone; S. Scialò A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method, Finite Elements in Analysis and Design, Volume 109 (2016), pp. 23 -36 | DOI

[24] J.-P. Berenger A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., Volume 114 (1994) no. 2, pp. 185-200 | DOI | MR | Zbl

[25] A. Bermúdez; D. Gómez; P. Salgado Mathematical Models and Numerical Simulation in Electromagnetism, Unitext, 74, Springer, 2014 | MR | Zbl

[26] J. E. Bishop A displacement-based finite element formulation for general polyhedra using harmonic shape functions, Internat. J. Numer. Methods Engrg., Volume 97 (2014) no. 1, pp. 1-31 | DOI | MR | Zbl

[27] P. B. Bochev; J. M. Hyman Principles of mimetic discretizations of differential operators, Compatible spatial discretizations (IMA Vol. Math. Appl.), Volume 142, Springer, New York, 2006, pp. 89-119 | DOI | MR | Zbl

[28] A. Bonito; J.-L. Guermond; F. Luddens Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl., Volume 408 (2013) no. 2, pp. 498-512 | DOI | MR | Zbl

[29] A. Bonito; J.-L. Guermond; F. Luddens An interior penalty method with C 0 finite elements for the approximation of the Maxwell equations in heterogeneous media: convergence analysis with minimal regularity, ESAIM Math. Model. Numer. Anal., Volume 50 (2016) no. 5, pp. 1457-1489 | DOI | MR | Zbl

[30] J. H. Bramble; J. E. Pasciak A new approximation technique for div-curl systems, Math. Comp., Volume 73 (2004) no. 248, pp. 1739-1762 | DOI | MR | Zbl

[31] J. H. Bramble; J. E. Pasciak Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem, Math. Comp., Volume 77 (2008) no. 261, pp. 1-10 | DOI | MR | Zbl

[32] S. C. Brenner; J. Cui; F. Li; L.-Y. Sung A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem, Numer. Math., Volume 109 (2008) no. 4, pp. 509-533 | DOI | MR | Zbl

[33] S. C. Brenner; Q. Guan; Li-Y. Sung Some estimates for virtual element methods, Comput. Methods Appl. Math., Volume 17 (2017) no. 4, pp. 553-574 | DOI | MR | Zbl

[34] S. C. Brenner; L. R. Scott The mathematical theory of finite element methods, Texts in Applied Mathematics, 15, Springer, New York, 2008, xviii+397 pages | DOI | MR | Zbl

[35] S. C. Brenner; L. Sung Virtual Element Methods on meshes with small edges or faces, Math. Models Methods Appl. Sci., Volume 28 (2018), pp. 1291-1336 | DOI | MR | Zbl

[36] F. Brezzi; L. D. Marini Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg., Volume 253 (2013), pp. 455-462 | DOI | MR | Zbl

[37] A. Buffa; P. Ciarlet; E. Jamelot Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements, Numer. Math., Volume 113 (2009) no. 4, pp. 497-518 | DOI | MR | Zbl

[38] E. Cáceres; G. N. Gatica A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem, IMA J. Numer. Anal., Volume 37 (2017) no. 1, pp. 296-331 | DOI | MR | Zbl

[39] E. Cáceres; G. N. Gatica; F.A. Sequeira A mixed virtual element method for the Brinkman problem, Math. Models Methods Appl. Sci., Volume 27 (2017) no. 04, pp. 707-743 | DOI | MR | Zbl

[40] A. Cangiani; E.H. Georgoulis; T. Pryer; O.J. Sutton A posteriori error estimates for the virtual element method, Numer. Math., Volume 137 (2017) no. 4, pp. 857-893 | DOI | MR

[41] L. Chen; J. Huang Some error analysis on virtual element methods, CALCOLO., Volume 55:5 (2018) no. 1 | DOI | MR | Zbl

[42] C.H.A. Cheng; S. Shkoller Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains, J. Math. Fluid. Mech., Volume 19 (2017), pp. 375-422 | DOI | MR | Zbl

[43] H. Chi; L. Beirão da Veiga; G.H. Paulino Some basic formulations of the virtual element method (VEM) for finite deformations, Comput. Methods Appl. Mech. Engrg., Volume 318 (2017), pp. 148 -192 | DOI | MR | Zbl

[44] S. H. Christiansen; A. Gillette Constructions of some minimal finite element systems, ESAIM Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 833-850 | DOI | MR | Zbl

[45] B. Cockburn; D. Di Pietro; A. Ern Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., Volume 50 (2016), pp. 635-650 | DOI | MR | Zbl

[46] B. Cockburn; J. Gopalakrishnan; R. Lazarov Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 1319-1365 | DOI | MR | Zbl

[47] M. Costabel A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl., Volume 157 (1991) no. 2, pp. 527-541 | DOI | MR | Zbl

[48] M. Costabel; M. Dauge; S. Nicaise Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal., Volume 33 (1999) no. 3, pp. 627-649 | DOI | Numdam | MR | Zbl

[49] F. Dassi; L. Mascotto Exploring High-order three dimensional Virtual Elements: bases and stabilizations, Comput. Math. Appl. (2018) no. 9, pp. 3379-3401 | DOI | MR | Zbl

[50] L. Demkowicz; J. Kurtz; D. Pardo; M. Paszenski; W. Rachowicz; A. Zdunek Computing with hp-adaptive finite elements. Vol. 2. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, Applied Mathematics and Nonlinear Science, Chapman & Hall/CRC, Boca Raton, 2008, xvi+417 pages

[51] V. Dolejší; M. Feistauer Discontinuous Galerkin method. Analysis and applications to compressible flow, Springer Series in Computational Mathematics, 48, Springer, Cham, 2015, xiv+572 pages | Zbl

[52] J. Droniou; R. Eymard; T. Gallouët; R. Herbin Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 13, pp. 2395-2432 | DOI | MR | Zbl

[53] H.-Y. Duan; R. C. E. Tan; S.-Y. Yang; C.-S. You Computation of Maxwell singular solution by nodal-continuous elements, J. Comput. Phys., Volume 268 (2014), pp. 63-83 | DOI | MR | Zbl

[54] H.-Y. Duan; R. C. E. Tan; S.-Y. Yang; C.-S. You A mixed H 1 -conforming finite element method for solving Maxwell’s equations with non-H 1 solution, SIAM J. Sci. Comput., Volume 40 (2018) no. 1, p. A224-A250 | DOI | MR | Zbl

[55] M. S. Floater Generalized barycentric coordinates and applications, Acta Numer., Volume 24 (2015), pp. 215-258 | DOI | MR | Zbl

[56] A. L. Gain; C. Talischi; G. H. Paulino On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg., Volume 282 (2014), pp. 132-160 | DOI | MR | Zbl

[57] A.L. Gain; G. H. Paulino; S. D. Leonardo; I. F. M. Menezes Topology optimization using polytopes, Comput. Methods Appl. Mech. Engrg., Volume 293 (2015), pp. 411-430 | DOI | MR | Zbl

[58] K. Gerdes A summary of infinite element formulations for exterior Helmholtz problems, Comput. Methods Appl. Mech. Engrg., Volume 164 (1998) no. 1-2, pp. 95-105 | DOI | MR | Zbl

[59] P. Houston; I. Perugia; D. Schötzau Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal., Volume 42 (2004) no. 1, pp. 434-459 | DOI | MR | Zbl

[60] S. R. Idelsohn; E. Oñate; N. Calvo; F. Del Pin The meshless finite element method, Internat. J. Numer. Methods Engrg., Volume 58 (2003) no. 6, pp. 893-912 | DOI | MR | Zbl

[61] H. Kanayama; R. Motoyama; K. Endo; F. Kikuchi Three dimensional Magnetostatic Analysis using Nédélec’s Elements, IEEE Trans. Magn., Volume 26 (1990), pp. 682-685 | DOI

[62] K. Lipnikov; G. Manzini; M. Shashkov Mimetic finite difference method, J. Comput. Phys., Volume 257 (2014) no. part B, pp. 1163-1227 | DOI | MR | Zbl

[63] L. Moheit; S. Marburg Infinite elements and their influence on normal and radiation modes in exterior acoustics, J. Comput. Acoust., Volume 25 (2017) no. 4, 1650020, 20 pages | DOI | MR

[64] P. Monk Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003, xiv+450 pages | Zbl

[65] D. Mora; G. Rivera; R. Rodríguez A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 8, pp. 1421-1445 | DOI | MR | Zbl

[66] S. Natarajan; S. Bordas; E.T. Ooi Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods, Internat. J. Numer. Methods Engrg., Volume 104 (2015) no. 13, pp. 1173-1199 | DOI | MR | Zbl

[67] A. Ortiz-Bernardin; A. Russo; N. Sukumar Consistent and stable meshfree Galerkin methods using the virtual element decomposition, Internat. J. Numer. Methods Engrg., Volume 112 (2017) no. 7, pp. 655-684 | DOI | MR

[68] I. Perugia; P. Pietra; A. Russo A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 783-808 | DOI | MR | Zbl

[69] S. Rjasanow; S. Weisser FEM with Trefftz trial functions on polyhedral elements, J. Comput. Appl. Math., Volume 263 (2014), pp. 202-217 | DOI | MR | Zbl

[70] N. Sukumar; E. A. Malsch Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Engrg., Volume 13 (2006) no. 1, pp. 129-163 | DOI | MR | Zbl

[71] G. Vacca Virtual element methods for hyperbolic problems on polygonal meshes, Comput. Math. Appl., Volume 74 (2017) no. 5, pp. 882-898 | DOI | MR | Zbl

[72] P. Wriggers; W.T. Rust; B.D. Reddy A virtual element method for contact, Comput. Mech., Volume 58 (2016) no. 6, pp. 1039-1050 | DOI | MR

[73] P. Wriggers; W.T. Rust; B.D. Reddy; B. Hudobivnik Efficient virtual element formulations for compressible and incompressible finite deformations, Comput. Mech., Volume 60 (2017) no. 2, pp. 253-268 | DOI | MR | Zbl

[74] J. Zhao; S. Chen; B. Zhang The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci., Volume 26 (2016) no. 09, pp. 1671-1687 | DOI | MR | Zbl

Cited by Sources: