Two-grid virtual element discretization of quasilinear elliptic problem
Abstract
In this paper a two grid algorithm for quasilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a quasilinear elliptic problem on a fine grid is reduced to the solution of a quasilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimate in H1 norm is derived. Numerical experiments are carried out to illustrate the theoretical findings.
Keyword : virtual element method, two grid algorithm, a priori error estimate
How to Cite
Chen, F., Yang, M., & Zhou, Z. (2024). Two-grid virtual element discretization of quasilinear elliptic problem. Mathematical Modelling and Analysis, 29(1), 77–89. https://doi.org/10.3846/mma.2024.17745
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Zhao, B. Zhang and X. Zhu. The nonconforming virtual element method for parabolic problems. Applied Numerical Mathematics, 143:97–111, 2019. https://doi.org/10.1016/j.apnum.2019.04.002
P.F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani. A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM Journal on Numerical Analysis, 52(1):386–404, 2014. https://doi.org/10.1137/13091141X
C. Bi and V. Ginting. Two-grid finite volume element method for linear and nonlinear elliptic problems. Numerische Mathematik, 108:177–198, 2007. https://doi.org/10.1007/s00211-007-0115-9
C. Bi and V. Ginting. Two-grid discontinuous galerkin method for quasilinear elliptic problems. Journal of Scientific Computing, 49:311–331, 2011. https://doi.org/10.1007/s10915-011-9463-9
M. Cai, M. Mu and J. Xu. Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach. SIAM Journal on Numerical Analysis, 47(5):3325–3338, 2009. https://doi.org/10.1137/080721868
A. Cangiani, P. Chatzipantelidis, G. Diwan and E.H. Georgoulis. Virtual element method for quasilinear elliptic problems. IMA Journal of Numerical Analysis, 40(4):2450–2472, 2020. https://doi.org/10.1093/imanum/drz035
A. Cangiani, G. Manzini and O.J. Sutton. Conforming and nonconforming virtual element methods for elliptic problems. IMA Journal of Numerical Analysis, 37(3):1317–1354, 2016. https://doi.org/10.1093/imanum/drw036
C. Chen, M. Yang and C. Bi. Two-grid methods for finite volume element approximations of nonlinear parabolic equations. Journal of Computational and Applied Mathematics, 228(1):123–132, 2009. https://doi.org/10.1016/j.cam.2008.09.001
L. Chen and Y. Chen. Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods. Journal of Scientific Computing, 49(3):383–401, 2011. https://doi.org/10.1007/s10915-011-9469-3
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01):199–214, 2013. https://doi.org/10.1142/S0218202512500492
L. Beirão da Veiga, C. Lovadina and G. Vacca. Virtual elements for the NavierStokes problem on polygonal meshes. SIAM Journal on Numerical Analysis, 56(3):1210–1242, 2018. https://doi.org/10.1137/17M1132811
J. Douglas Jr, T. Dupont and J. Serrin. Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Archive for Rational Mechanics and Analysis, 42(3):157–168, 1971. https://doi.org/10.1007/BF00250482
X. Liu, J. Li and Z. Chen. A nonconforming virtual element method for the Stokes problem on general meshes. Computer Methods in Applied Mechanics and Engineering, 320:694–711, 2017. https://doi.org/10.1016/j.cma.2017.03.027
C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes. Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Structural and Multidisciplinary Optimization, 45:309–328, 2012. https://doi.org/10.1007/s00158-011-0706-z
G. Vacca and L. Beirão da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numerical Methods for Partial Differential Equations, 31(6):2110–2134, 2015. https://doi.org/10.1002/num.21982
L. Wu and M.B. Allen. A two-grid method for mixed finite-element solution of reaction-diffusion equations. Numerical Methods for Partial Differential Equations, 15(3):317–332, 1999. https://doi.org/10.1002/(SICI)1098-2426(199905)15:3<317::AID-NUM4>3.0.CO;2-U
J. Xu. A novel two-grid method for semilinear elliptic equations. SIAM Journal on Scientific Computing, 15(1):231–237, 1994. https://doi.org/10.1137/0915016
J. Xu. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM Journal on Numerical Analysis, 33(5):1759–1777, 1996. https://doi.org/10.1137/S0036142992232949
J. Xu and A. Zhou. A two-grid discretization scheme for eigenvalue problems. Mathematics of Computation, 70(233):17–25, 2001. https://doi.org/10.1090/S0025-5718-99-01180-1
J. Zhao, B. Zhang and X. Zhu. The nonconforming virtual element method for parabolic problems. Applied Numerical Mathematics, 143:97–111, 2019. https://doi.org/10.1016/j.apnum.2019.04.002