A \(C^0\) virtual element method for the biharmonic eigenvalue problem. (English) Zbl 1480.65324
Summary: From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in [I. Babuška and J. Osborn, Handb. Numer. Anal. 2, 641–787 (1991; Zbl 0875.65087)] is \(h^{2k-2}\) for \(k \geq 2\). In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming \(H^1 (\Omega) \times H^1 (\Omega)\) formulation, following the variational formulation of Ciarlet-Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order \(h^2\) for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order.
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
Keywords:
virtual element method; biharmonic eigenvalue problem; spectral approximation; error analysisCitations:
Zbl 0875.65087References:
| [1] | Ahmad, B.; Alsaedi, A.; Brezzi, F.; Marini, L. D.; Russo, A., Equivalent projections for virtual element methods, Comput. Math. Appl., 66, 3, 376-391 (2013) · Zbl 1347.65172 |
| [2] | Andreev, A. B.; Lazarov, R. D.; Racheva, M. R., Postprocessing and higher order convergence of mixed finite element approximations of biharmonic eigenvalue problem, J. Comput. Appl. Math., 182, 2, 333-349 (2005) · Zbl 1075.65136 |
| [3] | Antonietti, P. F.; Beirão Da Veiga, L.; Scacchi, S.; Verani, M., A \(####\) virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54, 1, 34-56 (2016) · Zbl 1336.65160 |
| [4] | Argyris, J. H.; Fried, I.; Scharpf, D. W., The TUBA family of plate elements for the matrix displacement method, Aero. J. Roy. Aero. Soc., 72, 692, 701-709 (1968) |
| [5] | Artioli, E.; De Miranda, S.; Lovadina, C.; Patruno, L., A stress/displacement virtual element method for plane elasticity problems, Comput. Meth. Appl. Mech. Engrg., 325, 155-174 (2017) · Zbl 1439.74040 |
| [6] | Babska, I. and Osborn, J, Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991. |
| [7] | Beirão Da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 1, 199-214 (2013) · Zbl 1416.65433 |
| [8] | Beirão Da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., H(div) and H(curl)-conforming VEM, Numer. Math., 133, 303-332 (2015) · Zbl 1343.65133 |
| [9] | Beirão Da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., 50, 3, 727-747 (2016) · Zbl 1343.65134 |
| [10] | Beirão Da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., Virtual element methods for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26, 4, 729-750 (2016) · Zbl 1332.65162 |
| [11] | Beirão Da Veiga, L.; Dassi, F.; Russo, A., High-order virtual element method on polyhedral meshes, Comput. Math. Appl., 74, 5, 1110-1122 (2017) · Zbl 1448.65215 |
| [12] | Beirão Da Veiga, L.; Lovadina, C.; Vacca, G., Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal., 51, 2, 509-535 (2017) · Zbl 1398.76094 |
| [13] | Beirão Da Veiga, L.; Manzini, G., A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34, 2, 759-781 (2014) · Zbl 1293.65146 |
| [14] | Beirão Da Veiga, L.; Mora, D.; Rivera, G.; Rodríguez, R., A virtual element method for the acoustic vibration problem, Numer. Math., 136, 725-763 (2017) · Zbl 1397.76059 |
| [15] | Beirão Da Veiga, L.; Russo, A.; Vacca, G., The virtual element method with curved edges, ESAIM Math. Model. Numer. Anal., 53, 2, 375-404 (2019) · Zbl 1426.65163 |
| [16] | 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, 148-166 (2016) · Zbl 1351.76048 |
| [17] | Benedetto, M. F.; Berrone, S.; Borio, A.; Pieraccini, S.; Scialò, S., Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 311, 18-40 (2016) · Zbl 1439.76051 |
| [18] | Brenner, S. C.; Monk, P.; Sun, J., \(####\) interior penalty Galerkin method for biharmonic eigenvalue problems, Lect. Notes Comput. Sci. Eng., 106, 3-15 (2015) · Zbl 1349.65439 |
| [19] | Brenner, S.C. and Scott, R.L.. The Mathematical Theory of Finite Element Methods, Texts Appl. Math, Vol. 15, Springer-Verlag, New York, 2008. · Zbl 1135.65042 |
| [20] | Brezzi, F.; Falk, R. S.; Marini, L. D., Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48, 4, 1227-1240 (2014) · Zbl 1299.76130 |
| [21] | Cáceres, E.; Gatica, G. N., A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem, IMA J. Numer. Anal., 37, 1, 296-331 (2017) · Zbl 1433.76071 |
| [22] | Cáceres, E.; Gatica, G. N.; Sequeira, F. A., A mixed virtual element method for the Brinkman problem, Math. Models Methods Appl. Sci., 27, 4, 707-743 (2017) · Zbl 1432.65167 |
| [23] | Cangiani, A.; Gardini, F.; Manzini, G., Convergence of the mimetic finite difference method for eigenvalue problems in mixed form, Comput. Methods Appl. Mech. Engrg., 200, 9-12, 1150-1160 (2011) · Zbl 1225.65106 |
| [24] | Čertík, O.; Gardini, F.; Manzini, G.; Mascotto, L.; Vacca, G., The virtual element method for eigenvalue problems with potential terms on polytopic meshes, Appl. Math., 63, 3, 333-365 (2018) · Zbl 1488.65196 |
| [25] | Ciarlet, P.G., Basic error estimates for elliptic problems, Handbook of Numerical Analysis, Vol. 2, 1991, pp. 17-351, Available at doi:10.1016/S1570-8659(05)80039-0 · Zbl 0875.65086 |
| [26] | Ciarlet, P.G., Raviart, P.A., A mixed finite element method for the biharmonic equation, Math. Aspects Finite Elements Partial Diff. Equ. (1974), pp. 125-145. Available at https://doi.org/10.1016/B978-0-12-208350-1.50009-1 · Zbl 0337.65058 |
| [27] | Gardini, F.; Manzini, G.; Vacca, G., The nonconforming virtual element method for eigenvalue problems, ESAIM Math. Model. Numer. Anal., 53, 3, 749-774 (2019) · Zbl 1431.65214 |
| [28] | Gardini, F.; Vacca, G., Virtual element method for second order elliptic eigenvalue problems, IMA J. Numer. Anal., 38, 4, 2026-2054 (2017) · Zbl 1477.65182 |
| [29] | Ishihara, K., A mixed finite element method for the biharmonic eigenvalue problems of plate bending, Publ. Res. Inst. Math. Sci., 14, 2, 399-414 (1978) · Zbl 0389.73075 |
| [30] | Liu, X.; Chen, Z., A virtual element method for the Cahn-Hilliard problem in mixed form, Appl. Math. Lett., 87, 115-124 (2019) · Zbl 1410.65375 |
| [31] | Liu, X.; He, Z.; Chen, Z., A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form, Comput. Phys. Comm., 246 (2020) · Zbl 07678426 |
| [32] | Lü, T.; Liem, C.; Shih, T., A fourth order finite difference approximation to the eigenvalues of a clamped plate, J. Comput. Math., 6, 3, 267-271 (1988) · Zbl 0686.73055 |
| [33] | Monzón, G., A virtual element method for a biharmonic Steklov eigenvalue problem, Adv. Pure Appl. Math., 10, 4, 325-337 (2019) · Zbl 1430.31002 |
| [34] | Mora, D.; Rivera, G.; Rodríguez, R., A virtual element method for the Steklov eigenvalue, Math. Models Methods Appl. Sci., 25, 8, 1421-1445 (2015) · Zbl 1330.65172 |
| [35] | Mora, D.; Rivera, G.; Rodríguez, R., A posteriori error estimates for a virtual element method for the Steklov eigenvalue, Comput. Math. Appl., 74, 9, 2172-2190 (2017) · Zbl 1397.65246 |
| [36] | Mora, D.; Rivera, G.; Velásquez, I., A virtual element method for the vibration problem of Kirchhoff plates, ESAIM Math. Model. Numer. Anal., 52, 4, 1437-1456 (2018) · Zbl 1407.65274 |
| [37] | Mora, D.; Velásquez, I., A virtual element method for the transmission eigenvalue problem, Math. Models Methods Appl. Sci., 28, 14, 2803-2831 (2018) · Zbl 1411.65149 |
| [38] | Mora, D.; Velásquez, I., Virtual element for the buckling problem of Kirchhoff-Love plates, Comput. Methods Appl. Mech. Engrg., 360 (2020) · Zbl 1441.74266 |
| [39] | Sun, J.; Zhou, A., Finite Element Methods for Eigenvalue Problems (2016), CRC Press: CRC Press, Boca Raton, FL |
| [40] | Wang, L.; Xiong, C.; Wu, H.; Luo, F., A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems, Adv. Comput. Math., 45, 2623-2646 (2019) · Zbl 1432.65166 |
| [41] | Zhang, S.; Zhang, Z., Invalidity of decoupling a biharmonic equation to two poisson equations on non-convex polygons, Int. J. Numer. Anal. Model., 5, 1, 73-76 (2008) · Zbl 1136.35024 |
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