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Calcolo
Published in: Calcolo 2/2022

01-06-2022

Virtual element analysis of nonlocal coupled parabolic problems on polygonal meshes

Authors: M. Arrutselvi, D. Adak, E. Natarajan, S. Roy, S. Natarajan

Published in: Calcolo | Issue 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.
previous article Composite symmetric second derivative general linear methods for Hamiltonian systems
next article Componentwise perturbation analysis for the generalized Schur decomposition
Literature
1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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.
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.
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
Metadata
Title
Virtual element analysis of nonlocal coupled parabolic problems on polygonal meshes
Authors
M. Arrutselvi
D. Adak
E. Natarajan
S. Roy
S. Natarajan
Publication date
01-06-2022
Publisher
Springer International Publishing
Published in
Calcolo / Issue 2/2022
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-022-00459-4

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