# Article

Full entry | PDF   (1.1 MB)
Keywords:
Ritz-Volterra projection; stability; finite element; error estimates; initial- boundary-value problem; parabolic Volterra integro-differential equation
Summary:
In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some $L^\infty$ error estimates for finite element methods for parabolic integro-differential equations.
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, DOI 10.1007/BF02575943 | MR 1053754
[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. MR 1041253
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, 1978. MR 0520174 | 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. MR 0954953
[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. MR 1025089
[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. MR 1111453
[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. Соmр. Math., Vol. 1, No. 4 (1983) 376-383. MR 0726394
[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. MR 0527037 | Zbl 0401.65069
[10] R. Rannacher R. Scott: Some optimal error estimates for piecewise linear finite element approximations. Math. Соmр. 38 (1982) 437-445. MR 0645661
[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. MR 0562737
[12] R. Scott: Optimal $L^{\infty}$ estimates for the finite element on irregular meshes. Math. Соmр., 30 (1976) 681-697. MR 0436617 | Zbl 0349.65060
[13] V. Thomee N. Y. Zhang: Error estimates for semi-discrete finite element methods for parabolic integro-differential equations. Math. Соmр., 53 (1989) 121-139. MR 0969493
[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. MR 0351124

Partner of