黃宏財，博士論文，國立中山大學應用數學系，橢圓型方程的有限元整體超收斂，指導教授：李子才教授。

在這博士論文我們討論矩形元，Adini 元和 階Lagrange元，它們被建構在矩形有限空間。特殊矩形剖分能使有限元解 更易接近橢圓方程真解的插值 ，收斂率 比最優收斂高一階或二階。使用后處理我們構造高階有限元插值解 ，得到高階超收斂 。就我們所知，我們最早給Adini元及雙二次Lagrange元的后插值公式，且也是最早報告數值驗算對於Poisson方程在 模下超逼近四階至五階， ，整體超收斂五階， ，和對無窮模達到接近六階之高收斂率， 。

然而傳統有限元法對於奇異問題不有效，解精度很差。在這博士論文我們結合Ritz-Galerkin法與有限元法使用於奇異問題，如Motz問題。結合兩個方法僅在它們的共同邊界，我們採用簡易雜交，懲罰及懲罰加雜交技巧。在這博士論文理論分析得到幾乎最佳整體超收斂 ， ，當使用 階矩形元在光滑的子區域，及最佳整體超收斂3.5階， ，當使用Adini元在光滑的子區域。對於Ritz-Galerkin法與Adini元的結合數值驗算與理論分析完全吻合。

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.

0 Introduction
1 Adini’s Elements
1.1 Introduction
1.2 Adini’s Elements
1.3 Global Superconvergence
1.3.1 New error estimates
1.3.2 A posteriori interpolant formulas
1.3.3 Stability analysis
1.4 Proofs of Theorem 1.3.1
1.4.1 Preliminary Lemmas
1.4.2 Main proof
1.5 Numerical Experiments and Final Remarks
2 Biquadratic Lagrange Elements
2.1 Introduction
2.2 Biquadratic Lagrange Elements
2.3 Global Superconvergence
2.3.1 New error estimates
2.3.2 Proofs of Theorem 2.3.1
2.3.3 Proofs of Theorem 2.3.2
2.4 Numerical Experiments
2.4.1 Global superconvergence
2.4.2 Special case of h=k and f_xxyy=0
2.4.3 Comparisons
3 Simplified Hybrid Combinations of RGM and FEMs
4 Penalty plus Hybrid Combinations of RGM and FEMs
5 Applications to Landing Curves Problems
5.1 Introduction
5.2 Differential Equations and Variational Forms
5.3 Separation Techniques
5.3.1 Basic ideas
5.3.2 Linear boundary conditions
5.4 Nonlinear Blending Problems
5.4.1 Lagrange multiplier method for boundary nodal solutions
5.4.2 Hermite FEMs for linear ODEs
5.5 Fundamental Solutions
5.6 Error Analysis
5.7 Numerical Experiments
5.8 Concluding Remarks

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