# zbMATH — the first resource for mathematics

The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type. (English) Zbl 0732.65122
The Ritz-Volterra projection for the initial-boundary-value problem for the parabolic Volterra integro-differential equation $$u_ t+A(t)u+\int^{t}_{0}B(t,\tau)u(\tau)d\tau =f$$ was analyzed in detail by the first author, V. Thomeé and L. Wahlbin [SIAM J. Numer. Anal. 28, 1047-1070 (1991; Zbl 0728.65117)]. The present paper is concerned with the stability of this method and the derivation of $$L^{\infty}$$ error estimates for certain special choices of the operators A(t) and B(t,$$\tau$$).

##### MSC:
 65R20 Numerical methods for integral equations 45K05 Integro-partial differential equations
Full Text:
##### References:
 [1] J. R. Cannon, Yanping Lin: Non-classical $$H^1$$ projection and Galerkin methods for nonlinear parabolic integro-differential equations. Calcolo, 25 (1988) 187- 201, · Zbl 0685.65124 · doi:10.1007/BF02575943 [2] J. R. Cannon Y. Lin: A priori $$L^2$$ error estimates for finite element methods for nonlinear diffusion equations with memory. SJAM. J. Numer. Anal., 27 (1990) 595-607. · Zbl 0709.65122 · doi:10.1137/0727036 [3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, 1978. · Zbl 0383.65058 [4] E. Green-Yanik G. Fairweather: Finite element methods for parabolic and hyperbolic partial integro-differential equations. to appear in Nonlinear Analysis. · Zbl 0726.65160 [5] M. N. Le Roux V. Thomee: Numerical solution of semilinear integro-differential equations of parabolic type. SIAM J. Numer. Anal., 26 (1989) 1291-1309. · Zbl 0701.65091 · doi:10.1137/0726075 [6] Y. Lin V. Thomee L. Wahlbin: A Ritz-Volterra projection onto finite element spaces and application to integro and related equations. to appear in SIAM J. Numer. Anal. [7] Qun Lin, Tao Lu, Shu-min Shen: Maximum norm estimate, extrapolation and optimal points of stresses for the finite element methods on the strongly regular triangulalion. J. Comp. Math., Vol. 1, No. 4 (1983) 376-383. · Zbl 0563.65070 [8] Qun Lin, Qi-ding Zhou: Superconvergence Theory of Finite Element Methods. Book to appear. [9] J. A. Nitsche: $$L_{\infty}$$-convergence of finite element Galerkin approximations for parabolic problems. R.A.I.R.O., Vol. 13, No. 1, (1979) 31-51. · Zbl 0401.65069 · eudml:193332 [10] R. Rannacher R. Scott: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437-445. · Zbl 0483.65007 · doi:10.2307/2007280 [11] A. H. Schatz V. Thomée L. Wahlbin: Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pur. Appl. Math., XXXIII, (1980) 265-304. · Zbl 0414.65066 [12] R. Scott: Optimal $$L^{\infty}$$ estimates for the finite element on irregular meshes. Math. Comp., 30 (1976) 681-697. · Zbl 0349.65060 · doi:10.2307/2005390 [13] V. Thomee N. Y. Zhang: Error estimates for semi-discrete finite element methods for parabolic integro-differential equations. Math. Comp., 53 (1989) 121-139. [14] M. F. Wheeler: A priori $$L_2$$ error estimates for Galerkin methods to parabolic partial differential equations. SIAM J. Numer. Anal. 19 (1973) 723-759. · Zbl 0232.35060 · doi:10.1137/0710062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.