TY - JOUR

T1 - Superconvergence of Ritz-Galerkin finite element approximations for second order elliptic problems

AU - Wang, Chunmei

N1 - Funding Information:
The research of Chunmei Wang was partially supported by National Science Foundation Awards DMS-1522586 and DMS-1648171.
Publisher Copyright:
© 2017 Wiley Periodicals, Inc.

PY - 2018/5

Y1 - 2018/5

N2 - In this paper, the author derives an O(h4) -superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second-order elliptic equation - ∇.(A∇u)=f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A -equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

AB - In this paper, the author derives an O(h4) -superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second-order elliptic equation - ∇.(A∇u)=f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A -equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

KW - Euler-Maclaurin formula

KW - error estimate

KW - finite element method

KW - second-order elliptic problem

KW - superconvergence

UR - http://www.scopus.com/inward/record.url?scp=85044414376&partnerID=8YFLogxK

U2 - 10.1002/num.22231

DO - 10.1002/num.22231

M3 - Article

AN - SCOPUS:85044414376

SN - 0749-159X

VL - 34

SP - 838

EP - 856

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

IS - 3

ER -