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01.06.2022

Virtual element analysis of nonlocal coupled parabolic problems on polygonal meshes

verfasst von: M. Arrutselvi, D. Adak, E. Natarajan, S. Roy, S. Natarajan

Erschienen in: Calcolo | Ausgabe 2/2022

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Abstract

In this article, we consider the discretization of nonlocal coupled parabolic problem within the framework of the virtual element method. The presence of nonlocal coefficients not only makes the computation of the Jacobian more expensive in Newton’s method, but also destroys the sparsity of the Jacobian. In order to resolve this problem, an equivalent formulation that has very simple Jacobian is proposed. We derive the error estimates in the \(L^2\) and \(H^1\) norms. To further reduce the computational complexity, a linearized scheme without compromising the rate of convergence in different norms is proposed. Finally, the theoretical results are justified through numerical experiments over arbitrary polygonal meshes.

Vorheriger Artikel Composite symmetric second derivative general linear methods for Hamiltonian systems

Nächster Artikel Componentwise perturbation analysis for the generalized Schur decomposition

Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction–diffusion system modelling the spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173–184 (1981)MathSciNetMATHCrossRef

Bendahmane, M., Sepúlveda, M.A.: Convergence of finite volume scheme for nonlocal reaction diffusion systems modelling an epidemic disease. Discret. Contin. Dyn. Syst. Ser. B 11(4), 823–853 (2009)MathSciNetMATH

Xu, D., Zhao, X.-Q.: Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discret. Contin. Dyn. Syst. Ser. 4, 1043–1056 (2005)MathSciNetMATH

Shi, C., Roberts, G., Kiserow, D.: Effect of supercritical carbon dioxide on the diffusion coefficient of phenol in poly(bisphenol a carbonate). J. Polym. Sci. Part B 41, 1143–1156 (2003)CrossRef

Habib, S., Molina-Paris, C., Deisboeck, T.: Complex dynamics of tumors: modeling an emerging brain tumor system with coupled reaction–diffusion equations. Physica A 327, 501–524 (2003)MATHCrossRef

Raposo, C.A., Sepúlveda, M., Villagrán, O.V., Pereira, D.C., Santos, M.L.: Solution and asymptotic behaviour for a nonlocal coupled system of reaction–diffusion. Acta Applicandae Mathematicae 102(1), 37–56 (2008)MathSciNetMATHCrossRef

Chaudhary, S., Srivastava, V., Kumar, V.S., Srinivasan, B.: Finite element approximation of nonlocal parabolic problem. Numer. Methods Partial Differ. Equ. 33(3), 786–813 (2017)MathSciNetMATHCrossRef

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)MathSciNetMATHCrossRef

Beirão da Veiga, L., Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems, vol. 11. Springer, Berlin (2014)MATH

10.

Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113(3), 325–356 (2009)MathSciNetMATHCrossRef

11.

Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48(4), 1419–1443 (2010)MathSciNetMATHCrossRef

12.

Mu, L., Wang, J., Wei, G., Ye, X., Zhao, S.: Weak Galerkin methods for second order elliptic interface problems. J. Comput. Phys. 250, 106–125 (2013)MathSciNetMATHCrossRef

13.

Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)MathSciNetMATHCrossRef

14.

Sukumar, N., Malsch, E.A.: Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Eng. 13(1), 129–163 (2006)MathSciNetMATHCrossRef

15.

Sze, K., Sheng, N.: Polygonal finite element method for nonlinear constitutive modeling of polycrystalline ferroelectrics. Finite Elem. Anal. Des. 42(2), 107–129 (2005)CrossRef

16.

Bishop, J.: A displacement based finite element formulation for general polyhedra using harmonic shape functions. Int. J. Numer. Methods Eng. 97, 1–31 (2014)MathSciNetMATHCrossRef

17.

Natarajan, S., Ooi, E.T., Chiong, I., Song, C.: Convergence and accuracy of displacement based finite element formulation over arbitrary polygons: Laplace interpolants, strain smoothing and scaled boundary polygon formulation. Finite Elem. Anal. Des. 85, 101–122 (2014)MathSciNetCrossRef

18.

Ooi, E., Aaputra, A., Natarajan, S., Ooi, E., Song, C.: A dual scaled boundary finite element formulation over arbitrary faceted star convex polyhedra. Comput. Mech. 66, 27–47 (2020)MathSciNetMATHCrossRef

19.

Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)MathSciNetMATHCrossRef

20.

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)MathSciNetMATHCrossRef

21.

Vacca, G.: Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74, 882–898 (2017)MathSciNetMATHCrossRef

22.

Beirão da Veiga, L., Brezzi, F., Marini, L., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(08), 1541–1573 (2014)MathSciNetMATHCrossRef

23.

Floater, M.S., Lai, M.-J.: Polygonal spline spaces and the numerical solution of the Poisson equation. SIAM J Numer. Anal. 54, 794–827 (2016)MathSciNetCrossRef

24.

Sinu, A., Natarajan, S., Krishnapillai, S.: Quadratic serendipity finite elements over convex polyhedra. Int. J. Numer. Methods Eng. 113, 109–129 (2018)MathSciNetCrossRef

25.

Beirão da Veiga, L., Mora, D., Vacca, G.: The Stokes complex for virtual elements with application to Navier–Stokes flows. J. Sci. Comput. 81(2), 990–1018 (2019)MathSciNetMATHCrossRef

26.

Beirão da Veiga, L., Mora, D., Rivera, G.: Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates. Math. Comput. 88(315), 149–178 (2019)MathSciNetMATHCrossRef

27.

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)MathSciNetMATHCrossRef

28.

Adak, D., Natarajan, S.: Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique. Math. Comput. Simul. 172, 224–243 (2020)MathSciNetMATHCrossRef

29.

Adak, D., Natarajan, S.: Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes. Comput. Math. Appl. 79(10), 2858–2871 (2020)MathSciNetMATHCrossRef

30.

Cangiani, A., Chatzipantelidis, P., Diwan, G., Georgoulis, E.H.: Virtual element method for quasilinear elliptic problems. IMA J. Numer. Anal. 40(4), 2450–2472 (2020)MathSciNetMATHCrossRef

31.

Gardini, F., Vacca, G.: Virtual element method for second-order elliptic eigenvalue problems. IMA J. Numer. Anal. 38(4), 2026–2054 (2018)MathSciNetMATHCrossRef

32.

Gatica, G., Munar, M., Sequeira, F.: A mixed virtual element method for the Navier–Stokes equations. Math. Models Methods Appl. Sci. 28(14), 2719–2762 (2018)MathSciNetMATHCrossRef

33.

Cáceres, E., Gatica, G.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37(1), 296–331 (2017)MathSciNetMATHCrossRef

34.

Mascotto, L.: Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Meth. Partial Differ. Equ. 34(4), 1258–1281 (2018)MathSciNetMATHCrossRef

35.

Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)MathSciNetMATHCrossRef

36.

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)MathSciNetMATHCrossRef

37.

Gudi, T.: Finite element method for a nonlocal problem of Kirchhoff type. SIAM J. Numer. Anal. 50(2), 657–668 (2012)MathSciNetMATHCrossRef

38.

Chaudhary, S.: Finite element analysis of nonlocal coupled parabolic problem using Newton’s method. Comput. Math. Appl. 75(3), 981–1003 (2018)MathSciNetMATHCrossRef

39.

Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(01), 199–214 (2013)MathSciNetMATHCrossRef

40.

Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2016)MathSciNetMATH

41.

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)MathSciNetMATHCrossRef

42.

Beirão da Veiga, L., Dassi, F., Russo, A.: High-order virtual element method on polyhedral meshes. Comput. Math. Appl. 74(5), 1110–1122 (2017)MathSciNetMATHCrossRef

43.

Beirão da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(04), 729–750 (2016)MathSciNetMATHCrossRef

44.

Lions, J.L.: Quelques méthodes de résolution des problemes aux limites non linéaires, Dunod (1969)

45.

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 3. Springer, Berlin (2008)MATHCrossRef

46.

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)MathSciNetMATHCrossRef

Titel: Virtual element analysis of nonlocal coupled parabolic problems on polygonal meshes
verfasst von: M. Arrutselvi
D. Adak
E. Natarajan
S. Roy
S. Natarajan
Publikationsdatum: 01.06.2022
Verlag: Springer International Publishing
Erschienen in: Calcolo / Ausgabe 2/2022
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-022-00459-4

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Virtual element analysis of nonlocal coupled parabolic problems on polygonal meshes

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