Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Springer Science and Business Media LLC - Tập 104 - Trang 129-150 - 2006
Gabriel Acosta1, Gabriel Monzón1
1Universidad Nacional de General Sarmiento, Buenos Aires, Argentina

Tóm tắt

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p  ≥  3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p  ≥  1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p  ≥  3.

Tài liệu tham khảo

Acosta G., Durán R.G. (1999). The maximum angle condition for mixed and nonconforming elements: Application to the Stokes equations. SIAM J. Numer. Anal. 37, 18–36 Acosta G., Durán R.G. (2000). Error estimates for Q 1 isoparametric elements satisfying a weak angle condition. SIAM J. Numer. Anal. 38, 1073–1088 Acosta G., Durán R.G. (2004). An optimal Poincaré inequality in L 1 for convex domains. Proc. Am. Math. Soc. 132, 195–202 Apel, T.: Anisotropic finite elements: local estimates and applications. Advances in numerical mathematics. B. G. Teubner, Stuttgart, Leipzig (1999) Babuška I., Aziz A.K. (1976). On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226 Brenner S.C., Scott R.L.: The mathematical theory of finite element methods 2nd edn. Text in applied mathematics 15. Springer,Berlin Heidelberg Newyork (2002) Ciarlet P.G., Raviart P.A. (1972). Interpolation theory over curved elements, with applications to finite elements methods. Comput. Methods Appl. Mech. Eng. 1, 217–249 Jamet P. (1976). Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10, 46-61 Jamet P. (1977). Estimation of the interpolation error for quadrilateral finite elements which can degenerate into triangles. SIAM J. Numer. Anal. 14, 925–930 Ming P., Shi Z.C. (2002). Quadrilateral mesh revisited. Comput. Methods Appl. Mech. Eng. 191, 5671–5682 Payne L. E., Weinberger H.F. (1960). An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292 Verfürth R. (1999). Error estimates for some quasi-interpolation operator. AN Math. Model. Numer. Anal. 33, 695–713 Zenisek A., Vanmaele M. (1995). The interpolation theorem for narrow quadrilateral isoparametric finite elements. Numer. Math. 72, 123–141