Error Estimates of Conforming Virtual Element Methods with a Modified Symmetric Nitsche’s Formula for 2D Semilinear Parabolic Equations
Tóm tắt
In this paper, we study the
$$H^1$$
-conforming virtual element method for the spatial discretization of semilinear parabolic equations with inhomogeneous Dirichlet boundary conditions based on a modified symmetric Nitsche’s formula. Herein, the modified Nitsche’s formula is constructed by introducing a global lifting operator which maps the trace of an
$$H^1$$
-function into the global conforming virtual element space. In contrast to the classical symmetric Nitsche’s method, the penalty parameter in the modified symmetric Nitsche’s method does not need to be greater than a strictly positive lower bound, and it only needs to be greater than 0 to provide a coercive spatial bilinear form. For time discretization, the second-order backward difference formula is used. On the basis of some assumptions on the given problem data, the optimal error estimates in both an energy norm and
$$L^2$$
-norm are established for the semi-discrete and fully discrete schemes. The theoretical results are verified by some numerical experiments.
Từ khóa
Tài liệu tham khảo
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)
Vacca, G., Beirão da Veiga, L.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 31(6), 2110–2134 (2015)
Zhao, J., Zhang, B., Zhu, X.: The nonconforming virtual element method for parabolic problems. Appl. Numer. Math. 143, 97–111 (2019)
Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35(1), 222–245 (2019)
Adak, D., Natarajan, E., Kumar, S.: Virtual element method for semilinear hyperbolic problems on polygonal meshes. Int. J. Comput. Math. 96(5), 971–991 (2019)
Dehghan, M., Gharibi, Z., Eslahchi, M.R.: Unconditionally energy stable \(C^0\)-virtual element scheme for solving generalized Swift–Hohenberg equation. Appl. Numer. Math. 178, 304–328 (2022)
Li, M., Zhao, J., Wang, N., Chen, S.: Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: a unified framework. Comput. Methods Appl. Mech. Eng. 380, 113793 (2021)
Adak, D., Natarajan, S.: Virtual element methods for nonlocal parabolic problems on general type of meshes. Adv. Comput. Math. 46(5), 74 (2020)
Anaya, V., Bendahmane, M., Mora, D., Sepúlveda, M.: A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology. IMA J. Numer. Anal. 40(2), 1544–1576 (2020)
Gómez, S.A.: High-order interpolatory serendipity virtual element method for semilinear parabolic problems. Calcolo 59(3), 25 (2022)
Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Serendipity nodal VEM spaces. Comput. Fluids 141, 2–12 (2016)
Bürger, R., Kumar, S., Mora, D., Ruiz-Baier, R., Verma, N.: Virtual element methods for the three-field formulation of time-dependent linear poroelasticity. Adv. Comput. Math. 47(1), 2 (2021)
Adak, D., Natarajan, S.: On the \(H^1\) conforming virtual element method for time dependent Stokes equation. Math. Comput. Sci. 15(1), 135–154 (2021)
Adak, D., Mora, D., Natarajan, S., Silgado, A.: A virtual element discretization for the time dependent Navier–Stokes equations in stream-function formulation. ESAIM Math. Model. Numer. Anal. 55(5), 2535–2566 (2021)
Irisarri, D., Hauke, G.: Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations. Calcolo 56(4), 38 (2019)
Beirão da Veiga, L., Pichler, A., Vacca, G.: A virtual element method for the miscible displacement of incompressible fluids in porous media. Comput. Methods Appl. Mech. Eng. 375, 113649 (2021)
Beirão da Veiga, L., Dassi, F., Manzini, G., Mascotto, L.: Virtual elements for Maxwell’s equations. Comput. Math. Appl. 116, 82–99 (2022)
Alvarez, S.N., Bokil, V., Gyrya, V., Manzini, G.: The virtual element method for resistive magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 381, 113815 (2021)
Ben Belgacem, F., El Fekih, H., Raymond, J.P.: A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal. 34(2), 121–136 (2003)
Arada, N., Raymond, J.P.: Dirichlet boundary control of semilinear parabolic equations Part 1: problems with no state constraints. Appl. Math. Optim. 45(2), 125–143 (2002)
Bertoluzza, S., Pennacchio, M., Prada, D.: High order VEM on curved domains. Rend. Lincei Mat. Appl. 30(2), 391–412 (2019)
Bertoluzza, S., Pennacchio, M., Prada, D.: Weakly imposed Dirichlet boundary conditions for 2D and 3D virtual elements. Comput. Methods Appl. Mech. Eng. 400, 115454 (2022)
Cascavita, K.L., Chouly, F., Ern, A.: Hybrid high-order discretizations combined with Nitsche’s method for Dirichlet and Signorini boundary conditions. IMA J. Numer. Anal. 40(4), 2189–2226 (2020)
Burman, E.: A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50(4), 1959–1981 (2012)
Larson, M.G., Niklasson, A.J.: Analysis of a nonsymmetric discontinuous Galerkin method for elliptic problems: stability and energy error estimates. SIAM J. Numer. Anal. 42(1), 252–264 (2004)
Gudi, T., Nataraj, N., Pani, A.K.: On \(L^2\)-error estimate for nonsymmetric interior penalty Galerkin approximation to linear elliptic problems with nonhomogeneous Dirichlet data. J. Comput. Appl. Math. 228(1), 30–40 (2009)
Ludescher, T., Gross, S., Reusken, A.: A multigrid method for unfitted finite element discretizations of elliptic interface problems. SIAM J. Sci. Comput. 42(1), A318–A342 (2020)
Burman, E., Cicuttin, M., Delay, G., Ern, A.: An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems. SIAM J. Sci. Comput. 43(2), A859–A882 (2021)
Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer, Berlin (2003)
Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)
Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2017)
Brenner, S.C., Guan, Q., Sung, L.Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017)
Beirão da Veiga, L., 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(4), 729–750 (2016)
Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 5 (2018)
Mazzia, A.: A numerical study of the virtual element method in anisotropic diffusion problems. Math. Comput. Simul. 177, 63–85 (2020)
Atkinson, K., Han, W.: Theoretical Numerical Analysis. Springer, New York (2009)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)
Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)
Brazhnik, P.K., Tyson, J.J.: On traveling wave solutions of Fisher’s equation in two spatial dimensions. SIAM J. Appl. Math. 60(2), 371–391 (2000)
Tan, Y., Xu, H., Liao, S.J.: Explicit series solution of travelling waves with a front of Fisher equation. Chaos Solitons Fractals 31(2), 462–472 (2007)
Chafee, N., Infante, E.F.: A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4(1), 17–37 (1974)
Wu, F., Cheng, X., Li, D., Duan, J.: A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction–diffusion equations. Comput. Math. Appl. 75(8), 2835–2850 (2018)