A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

References

[1] H. Aminikhah and J. Biazar, A new analytical method for solving systems of Volterra integral equations, Int. J. Comput. Math. 87 (2010), 1142–1157. 10.1080/00207160903128497Suche in Google Scholar

[2] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. 10.1142/S0218202512500492Suche in Google Scholar

[3] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1541–1573. 10.1142/S021820251440003XSuche in Google Scholar

[4] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci. 26 (2016), no. 4, 729–750. 10.1142/S0218202516500160Suche in Google Scholar

[5] J. R. Cannon and Y. P. Lin, A priori L 2 error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal. 27 (1990), no. 3, 595–607. 10.1137/0727036Suche in Google Scholar

[6] M. Dehghan and Z. Gharibi, Virtual element method for solving an inhomogeneous Brusselator model with and without cross-diffusion in pattern formation, J. Sci. Comput. 89 (2021), no. 1, Paper No. 16. 10.1007/s10915-021-01626-5Suche in Google Scholar

[7] M. Dehghan and Z. Gharibi, A unified analysis of fully mixed virtual element method for wormhole propagation arising in the petroleum engineering, Comput. Math. Appl. 121 (2022), 30–51. 10.1016/j.camwa.2022.06.004Suche in Google Scholar

[8] M. Dehghan, Z. Gharibi and M. R. Eslahchi, Unconditionally energy stable C 0 -virtual element scheme for solving generalized Swift–Hohenberg equation, Appl. Numer. Math. 178 (2022), 304–328. 10.1016/j.apnum.2022.03.013Suche in Google Scholar

[9] M. Dehghan, Z. Gharibi and R. Ruiz-Baier, Optimal error estimates of coupled and divergence-free virtual element methods for the Poisson–Nernst–Planck/Navier–Stokes equations and applications in electrochemical systems, J. Sci. Comput. 94 (2023), no. 3, Paper No. 72. 10.1007/s10915-023-02126-4Suche in Google Scholar

[10] M. Dehghan and F. Shakeri, Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational iteration technique, Int. J. Numer. Methods Biomed. Eng. 26 (2010), no. 6, 705–715. 10.1002/cnm.1166Suche in Google Scholar

[11] B. Deka and R. C. Deka, A priori L ∞ ⁢ ( L 2 ) error estimates for finite element approximations to parabolic integro-differential equations with discontinuous coefficients, Proc. Indian Acad. Sci. Math. Sci. 129 (2019), no. 4, Paper No. 49. 10.1007/s12044-019-0494-8Suche in Google Scholar

[12] D. A. Di Pietro and R. Tittarelli, An introduction to hybrid high-order methods, Numerical Methods for PDEs, SEMA SIMAI Springer Ser. 15, Springer, Cham (2018), 75–128. 10.1007/978-3-319-94676-4_4Suche in Google Scholar

[13] J. Droniou, The Gradient Discretisation Method, Springer, Cham, 2018. 10.1007/978-3-319-79042-8Suche in Google Scholar

[14] F. Fakhar-Izadi and M. Dehghan, An efficient pseudo-spectral Legendre–Galerkin method for solving a nonlinear partial integro-differential equation arising in population dynamics, Math. Methods Appl. Sci. 36 (2013), no. 12, 1485–1511. 10.1002/mma.2698Suche in Google Scholar

[15] F. Fakhar-Izadi and M. Dehghan, Space-time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains, Comput. Math. Appl. 67 (2014), no. 10, 1884–1904. 10.1016/j.camwa.2014.03.016Suche in Google Scholar

[16] F. Fakhar-Izadi and M. Dehghan, Fully spectral collocation method for nonlinear parabolic partial integro-differential equations, Appl. Numer. Math. 123 (2018), 99–120. 10.1016/j.apnum.2017.08.007Suche in Google Scholar

[17] D. Goswami, A. K. Pani and S. Yadav, Optimal error estimates of two mixed finite element methods for parabolic integro-differential equations with nonsmooth initial data, J. Sci. Comput. 56 (2013), no. 1, 131–164. 10.1007/s10915-012-9666-8Suche in Google Scholar

[18] K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method, J. Comput. Phys. 257 (2014), 1163–1227. 10.1016/j.jcp.2013.07.031Suche in Google Scholar

[19] S.-O. Londen and O. J. Staffans, Volterra Equations, Lecture Notes in Math. 737. Springer, Berlin, (1979). 10.1007/BFb0064489Suche in Google Scholar

[20] R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35 (1977/78), no. 1, 1–19. 10.1090/qam/452184Suche in Google Scholar

[21] J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci. 184 (2003), no. 2, 201–222. 10.1016/S0025-5564(03)00041-5Suche in Google Scholar

[22] A. K. Pani and G. Fairweather, H 1 -Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 22 (2002), no. 2, 231–252. 10.1093/imanum/22.2.231Suche in Google Scholar

[23] A. K. Pani and T. E. Peterson, Finite element methods with numerical quadrature for parabolic integrodifferential equations, SIAM J. Numer. Anal. 33 (1996), no. 3, 1084–1105. 10.1137/0733053Suche in Google Scholar

[24] A. K. Pani and R. K. Sinha, Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data, Calcolo 37 (2000), no. 4, 181–205. 10.1007/s100920070001Suche in Google Scholar

[25] A. K. Pani and S. Yadav, An h ⁢ p -local discontinuous Galerkin method for parabolic integro-differential equations, J. Sci. Comput. 46 (2011), no. 1, 71–99. 10.1007/s10915-010-9384-zSuche in Google Scholar

[26] L. Perumal, A brief review on polygonal/polyhedral finite element methods, Math. Probl. Eng. 2018 (2018), Article ID 5792372. 10.1155/2018/5792372Suche in Google Scholar

[27] A. Radid and K. Rhofir, Partitioning differential transformation for solving integro-differential equations problem and application to electrical circuits, Math. Model. Eng. Probl. 6 (2019), 235–240. 10.18280/mmep.060211Suche in Google Scholar

[28] G. M. M. Reddy, A. B. Seitenfuss, D. d. O. Medeiros, L. Meacci, M. Assunção and M. Vynnycky, A compact FEM implementation for parabolic integro-differential equations in 2D, Algorithms 13 (2020), no. 10, Paper No. 242. 10.3390/a13100242Suche in Google Scholar

[29] F. Shakeri and M. Dehghan, A high order finite volume element method for solving elliptic partial integro-differential equations, Appl. Numer. Math. 65 (2013), 105–118. 10.1016/j.apnum.2012.10.002Suche in Google Scholar

[30] I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), no. 5, 1052–1061. 10.1137/0723073Suche in Google Scholar

[31] N. Sukumar and E. A. Malsch, Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Eng. 13 (2006), no. 1, 129–163. 10.1007/BF02905933Suche in Google Scholar

[32] G. Vacca and L. Beirão da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differential Equations 31 (2015), no. 6, 2110–2134. 10.1002/num.21982Suche in Google Scholar

[33] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp. 83 (2014), no. 289, 2101–2126. 10.1090/S0025-5718-2014-02852-4Suche in Google Scholar

[34] H. Yang, Superconvergence analysis of Galerkin method for semilinear parabolic integro-differential equation, Appl. Math. Lett. 128 (2022), Paper No. 107872. 10.1016/j.aml.2021.107872Suche in Google Scholar

[35] N. Y. Zhang, On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type, Math. Comp. 60 (1993), no. 201, 133–166. 10.1090/S0025-5718-1993-1149295-4Suche in Google Scholar