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Keywords:
conforming virtual element; eigenvalue problem; Hamiltonian equation; polygonal mesh
Summary:
We extend the conforming virtual element method (VEM) to the numerical resolution of eigenvalue problems with potential terms on a polytopic mesh. An important application is that of the Schrödinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provides a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the quantum harmonic oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[2] Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand Mathematical Studies 2, Princeton, Toronto (1965). MR 0178246 | Zbl 0142.37401
[3] Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L. D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013), 376-391. DOI 10.1016/j.camwa.2013.05.015 | MR 3073346 | Zbl 1347.65172
[4] Antonietti, P. F., Veiga, L. Beirão da, Scacchi, S., Verani, M.: A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54 (2016), 34-56. DOI 10.1137/15M1008117 | MR 3439765 | Zbl 1336.65160
[5] Antonietti, P. F., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28 (2018), 387-407. DOI 10.1142/S0218202518500100 | MR 3741104 | Zbl 1381.65090
[6] Antonietti, P. F., Mascotto, L., Verani, M.: A multigrid algorithm for the $p$-version of the virtual element method. ESAIM, Math. Model. Numer. Anal. 52 (2018), 337-364. DOI 10.1051/m2an/2018007 | MR 3808163
[7] Artioli, E., Miranda, S. De, Lovadina, C., Patruno, L.: A stress/displacement Virtual Element method for plane elasticity problems. Comput. Meth. Appl. Mech. Eng. 325 (2017), 155-174. DOI 10.1016/j.cma.2017.06.036 | MR 3693423
[8] Dios, B. Ayuso de, Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM, Math. Model. Numer. Anal. 50 (2016), 879-904. DOI 10.1051/m2an/2015090 | MR 3507277 | Zbl 1343.65140
[9] Babuška, I., Osborn, J.: Eigenvalue problems. Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1) P. G. Ciarlet North-Holland, Amsterdam (1991), 641-787. DOI 10.1016/s1570-8659(05)80042-0 | MR 1115240 | Zbl 0875.65087
[10] Bader, R. F. W.: A quantum theory of molecular structure and its applications. Chem. Rev. 91 (1991), 893-928. DOI 10.1021/cr00005a013
[11] Veiga, L. Beirão da, Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013), 199-214. DOI 10.1142/S0218202512500492 | MR 2997471 | Zbl 06144424
[12] Veiga, L. Beirão da, Brezzi, F., Dassi, F., Marini, L. D., Russo, A.: Virtual element approximation of 2D magnetostatic problems. Comput. Methods Appl. Mech. Eng. 327 (2017), 173-195. DOI 10.1016/j.cma.2017.08.013 | MR 3725767
[13] Veiga, L. Beirão da, Brezzi, F., Dassi, F., Marini, L. D., Russo, A.: Serendipity virtual elements for general elliptic equations in three dimensions. Chin. Ann. Math., Ser. B 39 (2018), 315-334. DOI 10.1007/s11401-018-1066-4 | MR 3757651 | Zbl 06877227
[14] Veiga, L. Beirão da, Brezzi, F., Marini, L. D., Russo, A.: The Hitchhiker's guide to the virtual element method. Math. Models Methods Appl. Sci. 24 (2014), 1541-1573. DOI 10.1142/S021820251440003X | MR 3200242 | Zbl 1291.65336
[15] Veiga, L. Beirão da, Brezzi, F., Marini, L. D., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016), 729-750. DOI 10.1142/S0218202516500160 | MR 3460621 | Zbl 1332.65162
[16] Veiga, L. Beirão da, Chernov, A., Mascotto, L., Russo, A.: Basic principles of $hp$ virtual elements on quasiuniform meshes. Math. Models Methods Appl. Sci. 26 (2016), 1567-1598. DOI 10.1142/S021820251650038X | MR 3509090 | Zbl 1344.65109
[17] Veiga, L. Beirão da, Dassi, F., Russo, A.: High-order virtual element method on polyhedral meshes. Comput. Math. Appl. 74 (2017), 1110-1122. DOI 10.1016/j.camwa.2017.03.021 | MR 3689939 | Zbl 06890717
[18] Veiga, L. Beirão da, Lipnikov, K., Manzini, G.: Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011), 1737-1760. DOI 10.1137/100807764 | MR 2837482 | Zbl 1242.65215
[19] Veiga, L. Beirão da, Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems. MS&A. Modeling, Simulation and Applications 11, Springer, Cham (2014). DOI 10.1007/978-3-319-02663-3 | MR 3135418 | Zbl 1286.65141
[20] Veiga, L. Beirão da, Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM, Math. Model. Numer. Anal. 51 (2017), 509-535. DOI 10.1051/m2an/2016032 | MR 3626409 | Zbl 06706760
[21] Veiga, L. Beirão da, Lovadina, C., Vacca, G.: Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56 (2018), 1210-1242. DOI 10.1137/17M1132811 | MR 3796371 | Zbl 06870040
[22] Veiga, L. Beirão da, Manzini, G.: A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014), 759-781. DOI 10.1093/imanum/drt018 | MR 3194807 | Zbl 1293.65146
[23] Veiga, L. Beirão da, Manzini, G.: Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM, Math. Model. Numer. Anal. 49 (2015), 577-599. DOI 10.1051/m2an/2014047 | MR 3342219 | Zbl 1346.65056
[24] Veiga, L. Beirão da, Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the acoustic vibration problem. Numer. Math. 136 (2017), 725-763. DOI 10.1007/s00211-016-0855-5 | MR 3660301 | Zbl 06751908
[25] Veiga, L. Beirão da, Russo, A., Vacca, G.: The virtual element method with curved edges. Available at https://arxiv.org/abs/1711.04306 29 pages (2017).
[26] Benedetto, M. F., Berrone, S., Borio, A., Pieraccini, S., Scialò, S.: A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. 306 (2016), 148-166. DOI 10.1016/j.jcp.2015.11.034 | MR 3432346 | Zbl 1351.76048
[27] Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numerica 19 (2010), 1-120. DOI 10.1017/S0962492910000012 | MR 2652780 | Zbl 1242.65110
[28] Brezzi, F., Marini, L. D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253 (2013), 455-462. DOI 10.1016/j.cma.2012.09.012 | MR 3002804 | Zbl 1297.74049
[29] Cáceres, E., Gatica, G. N.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37 (2017), 296-331. DOI 10.1016/j.camwa.2017.03.021 | MR 3614887
[30] Cai, Y., Bai, Z., Pask, J. E., Sukumar, N.: Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations. J. Comput. Phys. 255 (2013), 16-30. DOI 10.1016/j.jcp.2013.07.020 | MR 3109776 | Zbl 1349.81204
[31] Cangiani, A., Gardini, F., Manzini, G.: Convergence of the mimetic finite difference method for eigenvalue problems in mixed form. Comput. Methods Appl. Mech. Eng. 200 (2011), 1150-1160. DOI 10.1016/j.cma.2010.06.011 | MR 2796151 | Zbl 1225.65106
[32] Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54 (2016), 3411-3435. DOI 10.1137/15M1049531 | MR 3576570 | Zbl 06662515
[33] Cangiani, A., Manzini, G., Russo, A., Sukumar, N.: Hourglass stabilization and the virtual element method. Int. J. Numer. Meth. Eng. 102 (2015), 404-436. DOI 10.1002/nme.4854 | MR 3340083 | Zbl 1352.65475
[34] Cangiani, A., Manzini, G., Sutton, O.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. Analysis 37 (2017), 1317-1354. DOI 10.1093/imanum/drw036 | MR 3671497
[35] Chi, H., Veiga, L. Beirão da, Paulino, G. H.: Some basic formulations of the virtual element method (VEM) for finite deformations. Comput. Methods Appl. Mech. Eng. 318 (2017), 148-192. DOI 10.1016/j.cma.2016.12.020 | MR 3627175
[36] Dassi, F., Mascotto, L.: Exploring high-order three dimensional virtual elements: bases and stabilizations. Comput. Math. Appl. 75 (2018), 3379-3401. DOI 10.1016/j.camwa.2018.02.005 | MR 3785566
[37] Dauge, M.: Benchmark computations for Maxwell equations for the approximation of highly singular solutions. Available at\ https://perso.univ-rennes1.fr/monique.dauge/benchmax.html (2004).
[38] Ern, A., Guermond, J. L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences 159, Springer, New York (2004). DOI 10.1007/978-1-4757-4355-5 | MR 2050138 | Zbl 1059.65103
[39] Gardini, F., Manzini, G., Vacca, G.: The nonconforming virtual element method for eigenvalue problems. Available at https://arxiv.org/abs/1802.02942 (2018), 22 pages.
[40] Gardini, F., Vacca, G.: Virtual element method for second-order elliptic eigenvalue problems. (to appear) in IMA J. Numer. Anal. DOI 10.1093/imanum/drx063
[41] Grisvard, P.: Singularities in boundary value problems and exact controllability of hyperbolic systems. Optimization, Optimal Control and Partial Differential Equations V. Barbu et al. Internat. Ser. Numer. Math. 107, Birkhäuser, Basel (1992), 77-84. DOI 10.1007/978-3-0348-8625-3_8 | MR 1223360 | Zbl 0778.93007
[42] Gross, E. K. U., Dreizler, R. M.: Density Functional Theory. Springer Science & Business Media 337 (2013). DOI 10.1007/978-1-4757-9975-0 | MR 2743724
[43] Kato, T.: Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften 132, Springer, Berlin (1976). MR 0407617 | Zbl 0342.47009
[44] Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257 (2014), 1163-1227. DOI 10.1016/j.jcp.2013.07.031 | MR 3133437 | Zbl 1352.65420
[45] Mascotto, L., Perugia, I., Pichler, A.: Non-conforming harmonic virtual element method: $h$-and $p$-versions. Available at https://arxiv.org/abs/1801.00578 (2018), 27 pages. MR 3874797
[46] Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015), 1421-1445. DOI 10.1142/S0218202515500372 | MR 3340705 | Zbl 1330.65172
[47] Mora, D., Rivera, G., Rodríguez, R.: A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem. Comput. Math. Appl. 74 (2017), 2172-2190. DOI 10.1016/j.camwa.2017.05.016 | MR 3715326
[48] Mora, D., Rivera, G., Velásquez, I.: A virtual element method for the vibration problem of Kirchhoff plates. (to appear) in ESAIM Math. Model. Numer. Anal. DOI 10.1051/m2an/2017041
[49] Mora, D., Velásquez, I.: A virtual element method for the transmission eigenvalue problem. Available at https://arxiv.org/abs/1803.01979 (2018), 24 pages. MR 3895875
[50] Pask, J. E., Klein, B. M., Sterne, P. A., Fong, C. Y.: Finite-element methods in electronic-structure theory. Comput. Phys. Commun. 135 (2001), 1-34. DOI 10.1016/S0010-4655(00)00212-5 | MR 2700275 | Zbl 0984.81038
[51] Pask, J. E., Sterne, P. A.: Finite element methods in ab initio electronic structure calculations. Modelling Simul. Mater. Sci. Eng. 13 (2005), R71--R96. DOI 10.1088/0965-0393/13/3/R01
[52] Pask, J. E., Sukumar, N.: Partition of unity finite element method for quantum mechanical materials calculations. Extreme Mechanics Letters 11 (2017), 8-17. DOI 10.1016/j.eml.2016.11.003
[53] Pask, J. E., Sukumar, N., Guney, M., Hu, W.: Partition-of-unity finite-element method for large scale quantum molecular dynamics on massively parallel computational platforms. Technical report LLNL-TR-470692, Department of Energy LDRD (2011), Available at https://e-reports-ext.llnl.gov/pdf/471660.pdf\kern0pt
[54] Pickett, W. E.: Pseudopotential methods in condensed matter applications. Computer Physics Reports 9 (1989), 115-197. DOI 10.1016/0167-7977(89)90002-6
[55] Sukumar, N., Pask, J. E.: Classical and enriched finite element formulations for Bloch-periodic boundary conditions. Int. J. Numer. Methods Eng. 77 (2009), 1121-1138. DOI 10.1002/nme.2457 | MR 2490728 | Zbl 1156.81313
[56] Vacca, G.: Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74 (2017), 882-898. DOI 10.1016/j.camwa.2016.04.029 | MR 3689924
[57] Vacca, G.: An $H^1$-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28 (2018), 159-194. DOI 10.1142/S0218202518500057 | MR 3737081 | Zbl 06818909
[58] Wriggers, P., Rust, W. T., Reddy, B. D.: A virtual element method for contact. Comput. Mech. 58 (2016), 1039-1050. DOI 10.1007/s00466-016-1331-x | MR 3572918 | Zbl 06832903
[59] Yang, W., Ayers, P. W.: Density-functional theory. Computational Medicinal Chemistry for Drug Discovery CRC Press, Boca Raton (2003), 103-132.
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