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URN etd-0606103-220737 Author Hung-Tsai Huang Author's Email Address huanght@math.nsysu.edu.tw Statistics This thesis had been viewed 5249 times. Download 1871 times. Department Applied Mathematics Year 2002 Semester 2 Degree Ph.D. Type of Document Language English Title Global Superconvergence of Finite Element Methods for Elliptic Equations Date of Defense 2003-05-30 Page Count 180 Keyword global superconvergence Lagrange elements singularity combined methods posteriori interpolant Adini’s element finite element methods blending curves. elliptic equations Abstract In the dissertation we discuss the rectangular elements, Adini's elements and $p-$order Lagrange elements, which were constructed in the rectangular finite spaces. The special rectangular partitions enable the finite element solutions $u_h$ more efficient in interpolation of the true solution for Elliptic equation $u_I$. The convergence rates of $|u_h-u_I|_1$ are one or two orders higher than the optimal convergence rates. For post-processings we construct higher order interpolation operation $Pi_p$ to reach superconvergence $|u-Pi_p u_h|_1$. To our best knowledge, we at the first time provided the a posteriori interpolant formulas of Adini's elements and biquadratic Lagrange elements to obtain the global superconvergence, and at the first time reported the numerical verification for supercloseness $O(h^4)-O(h^5) $, global superconvergence $O(h^5)$ in $H^1$-norm and the high rates $O(h^6|ln h|)$ in the infinity norm for Poisson's equation(i.e., $-Delta u = f$).

Since the finite element methods is fail to deal with the singularity problems, in the dissertation, the combinations of the Ritz-Galerkin method and the finite element methods are used for the singularity problem, i.e., Motz's problem. To couple two methods along their common boundary, we adopt the simplified hybrid, penalty, and penalty plus hybrid techniques. The analysis are made in the dissertation to derive the almost best global superconvergence $O(h^{p+2-delta})$ in $H^1$-norm, $0<delta << 1$, for the combination using $p(geq 2)$-rectangles in the smooth subdomain, and the best global superconvergence $O(h^{3.5})$ in $H^1$-norm for combinations of Adini's elements in the smooth subdomain. The numerical experiments have been carried out for the combinations of the Ritz-Galerkin method and Adini's elements, to verify the theoretical superconvergence derived.Advisory Committee Weichung Wang - chair

Tzon-Tzer Lu - co-chair

Chang-Yi Wang - co-chair

Chun-Kong Law - co-chair

Chieh-Sen Huang - co-chair

Zi-Cai Li - advisor

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etd-0606103-220737.pdf Date of Submission 2003-06-06