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Non-classical \(H^ 1\) projection and Galerkin methods for non-linear parabolic integro-differential equations. (English) Zbl 0685.65124
The initial-boundary value problem for the equation \[ c(u)u_ t=\nabla \cdot \{a(u)\nabla u+\int^{t}_{0}b(x,t,r,u(x,r))\nabla u(x,r)dr\}+f(u\quad) \] is treated by Crank-Nicolson and extrapolated Crank-Nicolson approximations by using a non-classical \(H^ 1\) projection method. Optimal \(L^ 2\) error estimates are derived, and the schemes are shown to have second order accuracy in time. An advantage of the second scheme is that at each time-step only a linear algebraic system of equations is to be solved.
Reviewer: R.Gorenflo

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
Full Text: DOI
[1] H. Brunner,A survey of recent advances in the numerical treatment of Volterra integral and integral-differential equations, J. Comput. Appl. Math., 8 (1982), 76–102. · Zbl 0485.65087
[2] B. Budak, A. Pavlov,A difference method of solving boundary value problem for a quasi-linear integro-differential equation of parabolic type. Soviet math. Dokl. 14 (1975), 565–569. · Zbl 0301.65074
[3] J. R. Cannon, Xinkai Li, Yanping Lin,A Galerkin procedure for an integrodifferential equation of parabolic type, submitted.
[4] J. R. Cannon, Yanping Lin,A priori L 2 error estimates for Galerkin methods for nonlinea parabolic integro-differential, submitted. · Zbl 0709.65122
[5] P. G. Ciarlet,The finite Element Method for Elliptic Problems (1978), North-Holland. · Zbl 0383.65058
[6] J. Douglas Jr, B. Jones Jr,Numerical method for integral-differential equations of parabolic and hyperbolic type, Numer. Math., 4 (1962), 92–102.
[7] E. Greenwell-Yanik G. Fairweather,Finite element methods for parabolic and hyperbolic partial integro-differential equations, to appear in Nonlinear anal. · Zbl 0595.65122
[8] S. Londen, O. Staffans,Volterra Equations Lecture Notes in Math. 737, Sptring-Verlag, Berlin (1979).
[9] Marie-Noëlle Le Roux, Vidar Thoée,Numerical solution of a semilinear integrodifferential equation of parabolic type with nonsmooth data, to appear. · Zbl 0701.65091
[10] H. H. Rachford Jr.,Two level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations, SIAM, Numer. Anal. 10 (1973), 1010–1026. · Zbl 0268.65071
[11] I. Sloan, V. Tomée,Time discretization of an integral-differential equation of parabolic type, SIAM. numer. Anal., 23 (1986), 1052–1061. · Zbl 0608.65096
[12] M. F. Wheeler,A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM. Numer. Anal., 10 (1973), 723–759. · Zbl 0258.35041
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