Natural inner superconvergence for the finite element method. (English) Zbl 0611.65074

Zhu, Qiding

Finite element methods, Proc. China-France Symp., Beijing/China 1982, 935-960 (1983).

[For the entire collection see Zbl 0604.00019.]
Suppose \(u^ h\) is a Ritz-Galerkin approximation of the solution u of an elliptic boundary value problem, which is determined on the inner domain \(\Omega\subset {\mathcal D}\subset R^ N\). This paper obtains inner superconvergence estimates of error \(u-u^ h\) as follows: when r-1\(\geq 2\), in which r-1 is the degree of approximate polynomial, we have \[ | (u-u^ h)(x_ 0)| \leq C(h^{r+1}| \ln h| \| u\|_{r+1,\infty,\Omega}+\| u-u^ h\|_{-s}), \] where \(x_ 0\in \Omega_ 0\) is any node; when r-1\(\geq 1\), we have also estimates of the gradient: \[ | {\bar \nabla}(u-u^ h)(x_ 0)| \leq Ch^ r| \ln h| (\| u\|_{r+1,\infty,\Omega}+\| u\|_{3- \bar r,2,{\mathcal D}}), \] where \(x_ 0\in \Omega_ 0\) is any ”optimal point of stress”, and \(\Omega_ 0\subseteq \Omega\), \(\bar r=1\) if \(r=2\) or \(\bar r=0\) if \(r\geq 3\). Furthermore inner corresponding to \(L^ p\)- superconvergence estimates is obtained.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

finite element method; error estimates; Ritz-Galerkin; inner superconvergence; gradient; optimal point of stress

Citations:

Zbl 0604.00019