Journal of Integral Equations and Applications

Semi-Discrete Finite Element Approximations for Linear Parabolic Integro-Differential Equations with Integrable Kernels

Yanping Lin

Full-text: Open access

Article information

Source
J. Integral Equations Applications, Volume 10, Number 1 (1998), 51-83.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1181074208

Digital Object Identifier
doi:10.1216/jiea/1181074208

Mathematical Reviews number (MathSciNet)
MR1631540

Zentralblatt MATH identifier
0921.65095

Subjects
Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Secondary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Keywords
Integrable kernel finite element error estimates maximum norm superconvergence parabolic integro-differential

Citation

Lin, Yanping. Semi-Discrete Finite Element Approximations for Linear Parabolic Integro-Differential Equations with Integrable Kernels. J. Integral Equations Applications 10 (1998), no. 1, 51--83. doi:10.1216/jiea/1181074208. https://projecteuclid.org/euclid.jiea/1181074208


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References

  • W. Allegretto and Y. Lin, Numerical solutions for a class of differential equations in linear viscoelasticity, Calcolo 30 (1993), 69-88.
  • W. Allegretto, A. Zhou and Y. Lin, Long-time stability and error estimates for parabolic integro-differential equations with integrable kernels, Department of Mathematics, University of Alberta, 1995.
  • H. Brunner and P.J. van der Houwen, The numerical solution of Volterra equations, North-Holland, Amsterdam, 1986.
  • H. Brunner, J. Kauthen and A. Ostermann, Runge-Kutta time discretizations of parabolic Volterra integro-differential equations, J. Integral Equations Appl., to appear.
  • J.R. Cannon and Y. Lin, A priori $L^2$ error estimates for finite element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal. 27 1990), 595-607.
  • --------, Non-classical $H^1$ projection and Galerkin methods for nonlinear parabolic integro-differential equation, Calcolo 25 (1988), 187-201.
  • C. Chen, V. Thomee and L. Wahlbin, Finite element approximation of a parabolic integro-differential equations with a weakly singular kernel, Math. Comp. 58 (1992), 587-602.
  • P.G. Ciarlet, The finite element method for elliptic problems, North Holland, Amsterdam, 1978.
  • U. Jin Choi and R.C. MacCamy, Fractional order Volterra equations, in Volterra integrodifferential equations in Banach spaces and applications, Pitman Res. Notes Math. Ser. 190 (1989), 231-249.
  • Xu Da, On the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel I: Smooth initial data, Appl. Math. Comp. 58 (1993), 1-27.
  • G. Fairweather, Galerkin and collocation methods for partial integro-differential equations, in Integral equations and inverse problems, Pitman Res. Notes Math. Ser. 235 (1991), 76-85.
  • C.E. Greenwell-Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal. 12 (1988), 785-809.
  • U. Hornung and R. Showalter, Diffusion models for fractures media, J. Math. Anal. Appl. 147 (1990), 69-80.
  • J.P. Kauthen, Theoretical and computational aspects of continuous times collocation methods for Volterra-type integral and partial integro-differential equations, Ph.D. thesis, Universite de Fribourg, 1989.
  • M. Krizek and P. Neittaanmarki, On superconvergence techniques, Acta Appl. Math. 9 (1987), 175-198.
  • M.N. LeRoux and V. Thomee, Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal. 26 (1989), 1291-1309.
  • Q. Lin, T. Lu and S. Shen, Maximum norm estimates, extrapolation and optimal point of stress for the finite element methods on the strongly regular triangulation, J. Comp. Math. 1 (1983), 376-383.
  • Y. Lin, Galerkin methods for nonlinear parabolic integro-differential equations with nonlinear boundary conditions, SIAM J. Numer. Anal. 27 (1990), 608-621.
  • --------, Numerical solutions for integro-differential equations of parabolic type with weakly singular kernels, in Comparison methods and stability theory, Lecture Notes in Pure Appl. Math. 162 (1994), 261-268.
  • --------, On maximum norm estimates for Ritz-Volterra projections and applications to some time-dependent problems, J. Comp. Math. 15 (1997), 159-178.
  • Y. Lin, V. Thomee and L. Wahlbin, Ritz-Volterra projection onto finite element spaces and applications to integro-differential and related equations, SIAM J. Numer. Anal. 28 (1991), 1047-1070.
  • J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal. 27 (1990), 20-31.
  • J.A. Nitshe, $L_\infty$-convergence of finite element Galerkin approximations for parabolic problems, RAIRO 13 (1979), 31-51.
  • A. Pani and T. Peterson, The effect of numerical quadrature on semidiscrete finite element methods for parabolic integro-differential equations, SIAM J. Numer. Anal., to appear.
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer Verlag, New York, 1983.
  • M. Peszynska, Finite element approximation of differential equations with convolution term, Math. Comp. 65 (1996), 1019-1037.
  • R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 1-22.
  • M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical, England, 1987.
  • J.M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Math. 25 (1988), 319-327.
  • A.H. Schatz, V. Thomee and L. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math. 33 (1980), 265-304.
  • I.H. Sloan and V. Thomee, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), 1052-1061.
  • V. Thomee, Galerkin finite element methods for parabolic problems, Lecture Notes in Math. 1054 (1984).
  • V. Thomee and L. Wahlbin, Long-time numerical solution of a parabolic equation with memory, Math. Comp. 62 (1994), 477-496.
  • V. Thomee and N.Y. Zhang, Error estimates for semi-discrete finite element methods for parabolic integro-differential equations, Math. Comp. 53 (1989), 121-139.
  • M.F. Wheeler, A priori $L_2$ error estimates for Galerkin approximation to parabolic partial differential equations, SIAM J. Numer. Anal. 19 (1973), 723-759.
  • Qiding Zu and Qun Lin, Superconvergence theory for finite element methods, Hunan Scientific Press, Changsha, 1989.
  • Yi Yan and G. Fairweather, Orthogonal spline collocation methods for some partial integrodifferential equations, SIAM J. Numer. Anal. 29 (1992), 755-768.