TY - JOUR

T1 - Superconvergence in the generalized finite element method

AU - Babuška, Ivo

AU - Banerjee, Uday

AU - Osborn, John E.

PY - 2007/9/1

Y1 - 2007/9/1

N2 - In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945-981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727-758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289-314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct.

AB - In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945-981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727-758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289-314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct.

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U2 - 10.1007/s00211-007-0096-8

DO - 10.1007/s00211-007-0096-8

M3 - Article

AN - SCOPUS:34548107857

SN - 0029-599X

VL - 107

SP - 353

EP - 395

JO - Numerische Mathematik

JF - Numerische Mathematik

IS - 3

ER -