Global superconvergence of simplified hybrid combinations of the Ritz–Galerkin and FEMs for elliptic equations with singularities II. Lagrange elements and Adini's elements

Abstract

To solve the elliptic problems with singularities, the simplified hybrid combinations of the Ritz–Galerkin method and the finite element method (RGM–FEM) are explored, which lead to the high global superconvergence rates on the entire solution domain. Let the solution domain be split into a singular subdomain involving a singular point, and a regular subdomain where the true solution is smooth enough. In the singular subdomain, the singular particular functions are chosen to be admissible functions. In the regular subdomain either the k-order Lagrange rectangles or Adini's elements are adapted. Along their common boundary, the simplified hybrid techniques are employed to couple two different numerical methods. It is proven in this paper that the global superconvergence rates, O( h k+3/2 ), on the entire domain can be achieved for k(⩾2)-order Lagrange rectangles, and that the global superconvergence rates O( h 3.5) for the Adini's elements. Numerical experiments are reported for the combinations of the Ritz–Galerkin and Adini's methods. This paper presents a development of [Z.C. Li, Computing 65 (2000) 27–44] in high accurate solutions for the general case of the Poisson problems on a polygonal domain S estimates for the Sobolev norm ∥·∥ 1, given in a much more general sense than known before, cf. [P.G. Ciarlet, J.L. Lions (Eds.) Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991, pp. 17–351, 501–522; SIAM J. Sci. Statist. Comput. 11 (1990) 343; J. Comput. Appl. Math. 20 (1987) 341; Numer. Math. 63 (1992) 483; L. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995; Numer. Methods Partial Differential Equations 3 (1987) 65, 357].