Superconvergence of C0-Qk Finite Element Method for Elliptic Equations with Approximated Coefficients

We prove that the superconvergence of $C^0$-$Q^k$ finite element method at the Gauss Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $Q^k$ Lagrange interpolant at the Gauss Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $C^0$-$Q^2$ finite element method with $Q^2$ approximated coefficients.

[1]  Hao Li,et al.  Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations , 2019, J. Sci. Comput..

[2]  Mary F. Wheeler,et al.  An $L^\infty $ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials , 1974 .

[4]  J. J. Douglas,et al.  Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces , 1974 .

[5]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[6]  Zhimin Zhang,et al.  Ultraconvergence of high order FEMs for elliptic problems with variable coefficients , 2017, Numerische Mathematik.

[7]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[8]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .

[9]  Zhimin Zhang,et al.  2k superconvergence of Qk finite elements by anisotropic mesh approximation in weighted Sobolev spaces , 2017, Math. Comput..

[10]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[11]  M. Zlámal,et al.  Superconvergence of the gradient of finite element solutions , 1979 .

[12]  Chuanmiao Chen,et al.  The highest order superconvergence for bi-k degree rectangular elements at nodes: A proof of 2k-conjecture , 2012, Math. Comput..

[13]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[14]  Philippe G. Ciarlet,et al.  THE COMBINED EFFECT OF CURVED BOUNDARIES AND NUMERICAL INTEGRATION IN ISOPARAMETRIC FINITE ELEMENT METHODS , 1972 .

[15]  Inequalities for formally positive integro-differential forms , 1961 .

[16]  M. Bakker.  A note onCo Galerkin methods for two-point boundary problems , 1982 .

[17]  Franco Brezzi,et al.  On the numerical solution of plate bending problems by hybrid methods , 1975 .