Lagrange interpolation and finite element superconvergence. (English) Zbl 1042.65092

Li, Bo

Numer. Methods Partial Differ. Equations 20, No. 1, 33-59 (2004).

The author studies the supercloseness of a finite element solution of the Laplace equation with homogeneous Dirichlet boundary conditions in a \(d\)-dimensional domain, \(d\geq 1\), and the Lagrange interpolation of the solution of the continuous Laplace equation.
There are three main results contained in the paper. Firstly, it is proven that for the finite element spaces \(Q_k\), \(k\geq 1\), on \(d\)-dimensional tensor product meshes the interpolation points must be Lobatto points if the Lagrange interpolation and the finite element solution are superclose in the \(H^1\)-norm. This is the converse to well-known results. Secondly, for the finite element spaces \(P_k\), \(k\geq d+1\), \(d\geq 2\), on simplicial meshes, the standard Lagrange interpolation and the finite element solution are not superclose. The third result states that there are standard Lagrange interpolation points which are not superconvergence points of the finite element solution.

Reviewer: Volker John (Magdeburg)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

finite element method; Lagrange interpolation; superconvergence; Laplace equation

References:

[1]	and The mathematical theory of finite element methods, Springer-Verlag, New York, 1996.
[2]	The finite element method for elliptic problems. North-Holland, Amsterdam, 1978.
[3]	Chen, Xiangtan Univ J Nat Sci 1 (1978)
[4]	Andreev, Numer Methods Partial Differ Eq 4 pp 15– (1988)
[5]	Chen, Numer Math J Chinese Univ 2 pp 12– (1980)
[6]	Zhu, Xiangtan Univ J Nat Sci 3 pp 36– (1981)
[7]	Natural inner superconvergence for the finite element method, and editors, Proceedings of China-France symposium on finite element methods, Beijing, Gordon and Breach, New York, 1983, pp 935-960.
[8]	Chen, Numer Math J Chinese Univ 1 pp 73– (1979)
[9]	Superconvergence points in locally uniform finite element meshes for second order two-point boundary value problems. PhD thesis, Cornell University, 1996.
[10]	Douglas, RAIRO Mod?l Math Anal Num?r 8 pp 61– (1974)
[11]	Lesaint, RAIRO Mod?l Math Anal Num?r 13 pp 139– (1979)
[12]	Superconvergence in Galerkin finite element methods, Vol. 1605, Lecture notes in mathematics, Springer, Berlin, 1995. · Zbl 0826.65092
[13]	Zhang, Math Comp 65 pp 1431– (1996)
[14]	Chen, Xiangtan Univ J Natural Sci 3 pp 16– (1980)
[15]	and High accuracy theory of finite element methods. Huan Science Press, Hunan, China, 1995 (in Chinese).
[16]	Goodsell, Numer Methods Partial Differ Eq 10 pp 651– (1994)
[17]	Goodsell, Numer Methods Partial Differ Eq 7 pp 61– (1991)
[18]	Goodsell, Numer Methods Partial Differ Eq 7 pp 85– (1991)
[19]	K???ek, Acta Appl Math 9 pp 175– (1987)
[20]	Levine, IMA J Numer Anal 5 pp 407– (1985)
[21]	Lin, J Comp Math 3 pp 115– (1985)
[22]	Nakao, RAIRO Mod?l Math Anal Num?r 21 pp 679– (1987)
[23]	Zhu, Numer Math J Chinese Univ 3 pp 50– (1981)
[24]	Zhu, Appl Math 43 pp 401– (1998)
[25]	and Superconvergence theory of the finite element method, Hunan Science Press, Hunan, China, 1989 (in Chinese).
[26]	Brandts, IMA J Numer Anal
[27]	Li, Chinese J Numer Math Appl 12 pp 75– (1990)
[28]	Orthogonal polynomials, 3rd Ed., Amer Mathematics Soc, 1967.
[29]	Numerical quadrature and cubature. Academic Press, New York, 1980.
[30]	and Analysis of numerical methods. Dover, New York, 1994.