Finite element solution of a class of parabolic integro-differential equations with inhomogeneous jump conditions using FreeFEM++

Document Type : Research Paper


Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria.


The finite element solution of a class of parabolic integro–partial differential equations with interfaces is presented. The spatial discretization is based on the triangular element while a two-step implicit scheme together with the trapezoidal method is employed for time discretization. For the spatial discretization, the elements in the neighborhood of the interface are more refined such that the interface is at $\sigma$-distance from the approximate interface. The convergence rate of optimal order in L2-norm is analyzed with the assumption that the interface is arbitrary but smooth. Examples are given to support the theoretical findings with implementation on FreeFEM++.


  • [1] R. A. Adams, Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.
  • [2] M. O. Adewole, Almost optimal convergence of FEM-FDM for a linear parabolic interface problem, Electron. Trans. Numer. Anal., 46 (2017), 337–358.
  • [3] M. O. Adewole, FEM-IDS for a second order strongly damped wave equation with memory, Comput. Math. Appl., 80(12) (2020), 2896–2914.
  • [4] M. O. Adewole, Approximation of quasilinear hyperbolic problems with discontinuous coefficients: an optimal error estimate, Bull. Iranian Math. Soc., 47(2) (2021), 307–331.
  • [5] M. O. Adewole and V. F. Payne, Linearized four-step implicit scheme for nonlinear parabolic interface problems, Turkish J. Math., 42(6) (2018), 3034–3049.
  • [6] G. Amendola, M. Fabrizio, and J. M. Golden, Thermodynamics of materials with memory, Springer, Cham, second edition, [2021] ©2021. Theory and applications.
  • [7] I. Babuˇska, The finite element method for elliptic equations with discontinuous coefficients, Computing (Arch. Elektron. Rechnen), 5 (1970), 207–213.
  • [8] J. R. Cannon and Y. P. Lin, A priori L2 error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27(3) (1990), 595–607.
  • [9] V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl., 103(2) (1984), 575–588.
  • [10] C. Chen and T. Shih, Finite element methods for integrodifferential equations, volume 9 of Series on Applied Mathematics, World Scientific Publishing Co., Inc., River Edge, NJ, 1998.
  • [11] C. Chen, X. Zhang, G. Zhang, and Y. Zhang, A two-grid finite element method for nonlinear parabolic integrodifferential equations, Int. J. Comput. Math., 96(10) (2019), 2010–2023.
  • [12] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79(2) (1998), 175–202.
  • [13] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199–208.
  • [14] 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(6) (2010), 705–715.
  • [15] B. Deka and R. C. Deka, Finite element method for a class of parabolic integro-differential equations with interfaces, Indian J. Pure Appl. Math., 44(6) (2013), 823–847.
  • [16] B. Deka and R. C. Deka, A priori L∞(L2) error estimates for finite element approximations to parabolic integrodifferential equations with discontinuous coefficients, Proc. Indian Acad. Sci. Math. Sci., 129(4) (2019), 49.
  • [17] J. F. Epperson, An introduction to numerical methods and analysis, John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2013.
  • [18] 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(1) (2013), 131–164.
  • [19] H. Guo and H. Rui, Crank-Nicolson least-squares Galerkin procedures for parabolic integro-differential equations, Appl. Math. Comput., 180(2) (2006), 622–634.
  • [20] J. S. Gupta, R. K. Sinha, G. M. M. Reddy, and J. Jain, A posteriori error analysis of two-step backward differentiation formula finite element approximation for parabolic interface problems, J. Sci. Comput., 69(1) (2016), 406–429.
  • [21] J. S. Gupta, R. K. Sinha, G. M. M. Reddy, and J. Jain, New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems, Numer. Methods Partial Differential Equations, 33(2) (2017), 570–598.
  • [22] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31(2) (1968), 113–126.
  • [23] G. J. Habetler and R. L. Schiffman, A finite difference method for analyzing the compression of poro-viscoelastic media, Computing (Arch. Elektron. Rechnen), 6 (1970), 342–348.
  • [24] A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems Control Lett., 61(10) (2012), 999–1002.
  • [25] F. Hecht, New development in freefem++, J. Numer. Math., 20(3-4) (2012), 251–265.
  • [26] T. Hou, L. Chen, and Y. Yang, Two-grid methods for expanded mixed finite element approximations of semi-linear parabolic integro-differential equations, Appl. Numer. Math., 132 (2018), 163–181.
  • [27] Z. Jiang, L∞(L2) and L∞(L∞) error estimates for mixed methods for integro-differential equations of parabolic type, M2AN Math. Model. Numer. Anal., 33(3) (1999), 531–546.
  • [28] W. E. Kastenberg and P. L. Chambr´e, On the stability of nonlinear space dependent reactor kinetics, Nucl. Sci. Eng., 31 (1968), 67–79.
  • [29] R. B. Kellogg, Singularities in interface problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970), Academic Press, New York, 1971, 351–400.
  • [30] Y. P. Lin, V. Thom´ee, and L. B. Wahlbin, Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations, SIAM J. Numer. Anal., 28(4) (1991), 1047–1070.
  • [31] Y. Liu, H. Li, J. Wang, and W. Gao, A new positive definite expanded mixed finite element method for parabolic integrodifferential equations, J. Appl. Math., 24 (2012), 391372.
  • [32] R. Mohammadali, M. Bayareh, and A. Ahmadi Nadooshan, Numerical investigation on the effects of cell deformability and dld microfluidic device geometric parameters on the isolation of circulating tumor cells, Iranian Journal of Chemistry and Chemical Engineering, (2023), in press.
  • [33] A. K. Pani and G. Fairweather, H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 22(2) (2002), 231–252.
  • [34] A. K. Pani, V. Thom´ee, and L. B. Wahlbin, Numerical methods for hyperbolic and parabolic integro-differential equations, J. Integral Equations Appl., 4(4) (1992), 533–584.
  • [35] A. K. Pani and S. Yadav, An hp-local discontinuous Galerkin method for parabolic integro-differential equations, J. Sci. Comput., 46(1) (2011), 71–99.
  • [36] A. Patel, S. K. Acharya, and A. K. Pani, Stabilized Lagrange multiplier method for elliptic and parabolic interface problems, Appl. Numer. Math., 120 (2017), 287–304.
  • [37] D. Pradhan, N. Nataraj, and A. K. Pani, An explicit/implicit Galerkin domain decomposition procedure for parabolic integro-differential equations, J. Appl. Math. Comput., 28(1-2) (2008), 295–311.
  • [38] G. M. M. Reddy, Fully discrete a posteriori error estimates for parabolic integro-differential equations using the two-step backward differentiation formula, BIT Numerical Mathematics, 62(1) (2022), 251–277.
  • [39] G. M. M. Reddy, A. B. Seitenfuss, D. d. O. Medeiros, L. Meacci, M. Assun¸c˜ao, and M. Vynnycky, A compact FEM implementation for parabolic integro-differential equations in 2D, Algorithms (Basel), 13(10) (2020), 242.
  • [40] G. M. M. Reddy and R. K. Sinha, On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations, Math. Comp., 85(301) (2016), 2365–2390.
  • [41] G. M. M. Reddy, R. K. Sinha, and J. A. Cuminato, A posteriori error analysis of the Crank-Nicolson finite element method for parabolic integro-differential equations, J. Sci. Comput., 79(1) (2019), 414–441.
  • [42] M. A. Shallal, A. H. Taqi, H. N. Jabbar, H. Rezazadeh, B. F. Jumaa, A. Korkmaz, and A. Bekir, A numerical technique of the time fractional gas dynamics equation using finite element approach with cubic Hermit element, Appl. Comput. Math., 21(3) (2022), 269–278.
  • [43] N. Sharma and K. K. Sharma, Finite element method for a nonlinear parabolic integro-differential equation in higher spatial dimensions, Appl. Math. Model., 39(23-24) (2015), 7338–7350.
  • [44] R. K. Sinha, R. E. Ewing, and R. D. Lazarov, Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data, SIAM J. Numer. Anal., 47(5) (2009) 3269–3292.
  • [45] L. Song and C. Yang, Convergence of a second-order linearized BDF-IPDG for nonlinear parabolic equations with discontinuous coefficients, J. Sci. Comput., 70(2) (2017), 662–685.
  • [46] V. Thom´ee and N. Y. Zhang, Error estimates for semidiscrete finite element methods for parabolic integrodifferential equations, Math. Comp., 53(187) (1989), 121–139.
  • [47] R. Wait and A. R. Mitchell, Finite element analysis and applications, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985.
  • [48] H. Q. Wang, H. Li, S. He, and Y. Liu, Error estimate for a space-time discontinuous finite element method for a nonlinear parabolic integro-differential equation, Numer. Math. J. Chinese Univ., 35(2) (2013), 128–142.
  • [49] E. G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal., 12(8) (1988), 785–809.
  • [50] A. Zhu, Discontinuous mixed covolume methods for linear parabolic integrodifferential problems, J. Appl. Math., 8 (2014), 649468.
  • [51] A. Zhu, T. Xu, and Q. Xu, Weak Galerkin finite element methods for linear parabolic integro-differential equations, Numer. Methods Partial Differential Equations, 32(5) (2016), 1357–1377.
  • [52] A. L. Zhu, Z. W. Jiang, and Q. Xu, Expanded mixed covolume method for a linear integro-differential equation of parabolic type, Numer. Math. J. Chinese Univ., 31(3) (2009), 193–205.