The virtual element method for eigenvalue problems with potential terms on polytopic meshes

Ondřej Čertík; Francesca Gardini; Gianmarco Manzini; Giuseppe Vacca

Applications of Mathematics (2018)

  • Volume: 63, Issue: 3, page 333-365
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Čertík, Ondřej, et al. "The virtual element method for eigenvalue problems with potential terms on polytopic meshes." Applications of Mathematics 63.3 (2018): 333-365. <http://eudml.org/doc/294250>.

@article{Čertík2018,
abstract = {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.},
author = {Čertík, Ondřej, Gardini, Francesca, Manzini, Gianmarco, Vacca, Giuseppe},
journal = {Applications of Mathematics},
keywords = {conforming virtual element; eigenvalue problem; Hamiltonian equation; polygonal mesh},
language = {eng},
number = {3},
pages = {333-365},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The virtual element method for eigenvalue problems with potential terms on polytopic meshes},
url = {http://eudml.org/doc/294250},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Čertík, Ondřej
AU - Gardini, Francesca
AU - Manzini, Gianmarco
AU - Vacca, Giuseppe
TI - The virtual element method for eigenvalue problems with potential terms on polytopic meshes
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 333
EP - 365
AB - 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.
LA - eng
KW - conforming virtual element; eigenvalue problem; Hamiltonian equation; polygonal mesh
UR - http://eudml.org/doc/294250
ER -

References

top
  1. Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  2. Agmon, S., Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies 2, Princeton, Toronto (1965). (1965) Zbl0142.37401MR0178246
  3. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L. D., Russo, A., 10.1016/j.camwa.2013.05.015, Comput. Math. Appl. 66 (2013), 376-391. (2013) Zbl1347.65172MR3073346DOI10.1016/j.camwa.2013.05.015
  4. Antonietti, P. F., Veiga, L. Beirão da, Scacchi, S., Verani, M., 10.1137/15M1008117, SIAM J. Numer. Anal. 54 (2016), 34-56. (2016) Zbl1336.65160MR3439765DOI10.1137/15M1008117
  5. Antonietti, P. F., Manzini, G., Verani, M., 10.1142/S0218202518500100, Math. Models Methods Appl. Sci. 28 (2018), 387-407. (2018) Zbl1381.65090MR3741104DOI10.1142/S0218202518500100
  6. Antonietti, P. F., Mascotto, L., Verani, M., 10.1051/m2an/2018007, ESAIM, Math. Model. Numer. Anal. 52 (2018), 337-364. (2018) MR3808163DOI10.1051/m2an/2018007
  7. Artioli, E., Miranda, S. De, Lovadina, C., Patruno, L., 10.1016/j.cma.2017.06.036, Comput. Meth. Appl. Mech. Eng. 325 (2017), 155-174. (2017) MR3693423DOI10.1016/j.cma.2017.06.036
  8. Dios, B. Ayuso de, Lipnikov, K., Manzini, G., 10.1051/m2an/2015090, ESAIM, Math. Model. Numer. Anal. 50 (2016), 879-904. (2016) Zbl1343.65140MR3507277DOI10.1051/m2an/2015090
  9. Babuška, I., Osborn, J., 10.1016/s1570-8659(05)80042-0, Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1) P. G. Ciarlet North-Holland, Amsterdam (1991), 641-787. (1991) Zbl0875.65087MR1115240DOI10.1016/s1570-8659(05)80042-0
  10. Bader, R. F. W., 10.1021/cr00005a013, Chem. Rev. 91 (1991), 893-928. (1991) DOI10.1021/cr00005a013
  11. Veiga, L. Beirão da, Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., Russo, A., 10.1142/S0218202512500492, Math. Models Methods Appl. Sci. 23 (2013), 199-214. (2013) Zbl06144424MR2997471DOI10.1142/S0218202512500492
  12. Veiga, L. Beirão da, Brezzi, F., Dassi, F., Marini, L. D., Russo, A., 10.1016/j.cma.2017.08.013, Comput. Methods Appl. Mech. Eng. 327 (2017), 173-195. (2017) MR3725767DOI10.1016/j.cma.2017.08.013
  13. Veiga, L. Beirão da, Brezzi, F., Dassi, F., Marini, L. D., Russo, A., 10.1007/s11401-018-1066-4, Chin. Ann. Math., Ser. B 39 (2018), 315-334. (2018) Zbl06877227MR3757651DOI10.1007/s11401-018-1066-4
  14. Veiga, L. Beirão da, Brezzi, F., Marini, L. D., Russo, A., 10.1142/S021820251440003X, Math. Models Methods Appl. Sci. 24 (2014), 1541-1573. (2014) Zbl1291.65336MR3200242DOI10.1142/S021820251440003X
  15. Veiga, L. Beirão da, Brezzi, F., Marini, L. D., Russo, A., 10.1142/S0218202516500160, Math. Models Methods Appl. Sci. 26 (2016), 729-750. (2016) Zbl1332.65162MR3460621DOI10.1142/S0218202516500160
  16. Veiga, L. Beirão da, Chernov, A., Mascotto, L., Russo, A., 10.1142/S021820251650038X, Math. Models Methods Appl. Sci. 26 (2016), 1567-1598. (2016) Zbl1344.65109MR3509090DOI10.1142/S021820251650038X
  17. Veiga, L. Beirão da, Dassi, F., Russo, A., 10.1016/j.camwa.2017.03.021, Comput. Math. Appl. 74 (2017), 1110-1122. (2017) Zbl06890717MR3689939DOI10.1016/j.camwa.2017.03.021
  18. Veiga, L. Beirão da, Lipnikov, K., Manzini, G., 10.1137/100807764, SIAM J. Numer. Anal. 49 (2011), 1737-1760. (2011) Zbl1242.65215MR2837482DOI10.1137/100807764
  19. Veiga, L. Beirão da, Lipnikov, K., Manzini, G., 10.1007/978-3-319-02663-3, MS&A. Modeling, Simulation and Applications 11, Springer, Cham (2014). (2014) Zbl1286.65141MR3135418DOI10.1007/978-3-319-02663-3
  20. Veiga, L. Beirão da, Lovadina, C., Vacca, G., 10.1051/m2an/2016032, ESAIM, Math. Model. Numer. Anal. 51 (2017), 509-535. (2017) Zbl06706760MR3626409DOI10.1051/m2an/2016032
  21. Veiga, L. Beirão da, Lovadina, C., Vacca, G., 10.1137/17M1132811, SIAM J. Numer. Anal. 56 (2018), 1210-1242. (2018) Zbl06870040MR3796371DOI10.1137/17M1132811
  22. Veiga, L. Beirão da, Manzini, G., 10.1093/imanum/drt018, IMA J. Numer. Anal. 34 (2014), 759-781. (2014) Zbl1293.65146MR3194807DOI10.1093/imanum/drt018
  23. Veiga, L. Beirão da, Manzini, G., 10.1051/m2an/2014047, ESAIM, Math. Model. Numer. Anal. 49 (2015), 577-599. (2015) Zbl1346.65056MR3342219DOI10.1051/m2an/2014047
  24. Veiga, L. Beirão da, Mora, D., Rivera, G., Rodríguez, R., 10.1007/s00211-016-0855-5, Numer. Math. 136 (2017), 725-763. (2017) Zbl06751908MR3660301DOI10.1007/s00211-016-0855-5
  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). (2017) MR3939306
  26. Benedetto, M. F., Berrone, S., Borio, A., Pieraccini, S., Scialò, S., 10.1016/j.jcp.2015.11.034, J. Comput. Phys. 306 (2016), 148-166. (2016) Zbl1351.76048MR3432346DOI10.1016/j.jcp.2015.11.034
  27. Boffi, D., 10.1017/S0962492910000012, Acta Numerica 19 (2010), 1-120. (2010) Zbl1242.65110MR2652780DOI10.1017/S0962492910000012
  28. Brezzi, F., Marini, L. D., 10.1016/j.cma.2012.09.012, Comput. Methods Appl. Mech. Eng. 253 (2013), 455-462. (2013) Zbl1297.74049MR3002804DOI10.1016/j.cma.2012.09.012
  29. Cáceres, E., Gatica, G. N., 10.1016/j.camwa.2017.03.021, IMA J. Numer. Anal. 37 (2017), 296-331. (2017) MR3614887DOI10.1016/j.camwa.2017.03.021
  30. Cai, Y., Bai, Z., Pask, J. E., Sukumar, N., 10.1016/j.jcp.2013.07.020, J. Comput. Phys. 255 (2013), 16-30. (2013) Zbl1349.81204MR3109776DOI10.1016/j.jcp.2013.07.020
  31. Cangiani, A., Gardini, F., Manzini, G., 10.1016/j.cma.2010.06.011, Comput. Methods Appl. Mech. Eng. 200 (2011), 1150-1160. (2011) Zbl1225.65106MR2796151DOI10.1016/j.cma.2010.06.011
  32. Cangiani, A., Gyrya, V., Manzini, G., 10.1137/15M1049531, SIAM J. Numer. Anal. 54 (2016), 3411-3435. (2016) Zbl06662515MR3576570DOI10.1137/15M1049531
  33. Cangiani, A., Manzini, G., Russo, A., Sukumar, N., 10.1002/nme.4854, Int. J. Numer. Meth. Eng. 102 (2015), 404-436. (2015) Zbl1352.65475MR3340083DOI10.1002/nme.4854
  34. Cangiani, A., Manzini, G., Sutton, O., 10.1093/imanum/drw036, IMA J. Numer. Anal. Analysis 37 (2017), 1317-1354. (2017) MR3671497DOI10.1093/imanum/drw036
  35. Chi, H., Veiga, L. Beirão da, Paulino, G. H., 10.1016/j.cma.2016.12.020, Comput. Methods Appl. Mech. Eng. 318 (2017), 148-192. (2017) MR3627175DOI10.1016/j.cma.2016.12.020
  36. Dassi, F., Mascotto, L., 10.1016/j.camwa.2018.02.005, Comput. Math. Appl. 75 (2018), 3379-3401. (2018) MR3785566DOI10.1016/j.camwa.2018.02.005
  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). (2004) 
  38. Ern, A., Guermond, J. L., 10.1007/978-1-4757-4355-5, Applied Mathematical Sciences 159, Springer, New York (2004). (2004) Zbl1059.65103MR2050138DOI10.1007/978-1-4757-4355-5
  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. (2018) MR3959470
  40. Gardini, F., Vacca, G., 10.1093/imanum/drx063, (to appear) in IMA J. Numer. Anal. DOI10.1093/imanum/drx063
  41. Grisvard, P., 10.1007/978-3-0348-8625-3_8, Optimization, Optimal Control and Partial Differential Equations V. Barbu et al. Internat. Ser. Numer. Math. 107, Birkhäuser, Basel (1992), 77-84. (1992) Zbl0778.93007MR1223360DOI10.1007/978-3-0348-8625-3_8
  42. Gross, E. K. U., Dreizler, R. M., 10.1007/978-1-4757-9975-0, Springer Science & Business Media 337 (2013). (2013) MR2743724DOI10.1007/978-1-4757-9975-0
  43. Kato, T., Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften 132, Springer, Berlin (1976). (1976) Zbl0342.47009MR0407617
  44. Lipnikov, K., Manzini, G., Shashkov, M., 10.1016/j.jcp.2013.07.031, J. Comput. Phys. 257 (2014), 1163-1227. (2014) Zbl1352.65420MR3133437DOI10.1016/j.jcp.2013.07.031
  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. (2018) MR3874797
  46. Mora, D., Rivera, G., Rodríguez, R., 10.1142/S0218202515500372, Math. Models Methods Appl. Sci. 25 (2015), 1421-1445. (2015) Zbl1330.65172MR3340705DOI10.1142/S0218202515500372
  47. Mora, D., Rivera, G., Rodríguez, R., 10.1016/j.camwa.2017.05.016, Comput. Math. Appl. 74 (2017), 2172-2190. (2017) MR3715326DOI10.1016/j.camwa.2017.05.016
  48. Mora, D., Rivera, G., Velásquez, I., 10.1051/m2an/2017041, (to appear) in ESAIM Math. Model. Numer. Anal. DOI10.1051/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. (2018) MR3895875
  50. Pask, J. E., Klein, B. M., Sterne, P. A., Fong, C. Y., 10.1016/S0010-4655(00)00212-5, Comput. Phys. Commun. 135 (2001), 1-34. (2001) Zbl0984.81038MR2700275DOI10.1016/S0010-4655(00)00212-5
  51. Pask, J. E., Sterne, P. A., 10.1088/0965-0393/13/3/R01, Modelling Simul. Mater. Sci. Eng. 13 (2005), R71--R96. (2005) DOI10.1088/0965-0393/13/3/R01
  52. Pask, J. E., Sukumar, N., 10.1016/j.eml.2016.11.003, Extreme Mechanics Letters 11 (2017), 8-17. (2017) DOI10.1016/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. (2011) 
  54. Pickett, W. E., 10.1016/0167-7977(89)90002-6, Computer Physics Reports 9 (1989), 115-197. (1989) DOI10.1016/0167-7977(89)90002-6
  55. Sukumar, N., Pask, J. E., 10.1002/nme.2457, Int. J. Numer. Methods Eng. 77 (2009), 1121-1138. (2009) Zbl1156.81313MR2490728DOI10.1002/nme.2457
  56. Vacca, G., 10.1016/j.camwa.2016.04.029, Comput. Math. Appl. 74 (2017), 882-898. (2017) MR3689924DOI10.1016/j.camwa.2016.04.029
  57. Vacca, G., 10.1142/S0218202518500057, Math. Models Methods Appl. Sci. 28 (2018), 159-194. (2018) Zbl06818909MR3737081DOI10.1142/S0218202518500057
  58. Wriggers, P., Rust, W. T., Reddy, B. D., 10.1007/s00466-016-1331-x, Comput. Mech. 58 (2016), 1039-1050. (2016) Zbl06832903MR3572918DOI10.1007/s00466-016-1331-x
  59. Yang, W., Ayers, P. W., Density-functional theory, Computational Medicinal Chemistry for Drug Discovery CRC Press, Boca Raton (2003), 103-132. (2003) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.