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Diagram

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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$$ \begin{align*} a &= b & c &= d & e &= f \\\ a’ &= b’ & c’ &= d’ & e’ &= f' \end{align*} $$

$$ \hat u_0(x_k):=\left\{ \begin{align*} &g(\bar x_k), & x_k \in \partial \Omega_{\epsilon} \\\ &\hat u_0 (x_{k+1}), & \text{otherwise} \end{align*} \right. $$

$$ \begin{array}{rl} \biggl\{ \frac{1}{g(x)} \biggr\}^{\prime} &=& \lim_{h \to 0}\frac{\frac{1}{g(x+h)}-\frac{1}{g(x)}}{h} \cr\cr &=& \lim_{h \to 0}\frac{g(x)-g(x+h)}{g(x) g(x+h) h} \cr\cr &=& -\frac{1}{ \bigl\{ g(x) \bigr\}^2} \lim_{h \to 0}\frac{g(x+h)-g(x)}{h} \cr\cr &=& -\frac{g^{\prime}(x)}{ \bigl\{g(x) \bigr\}^2} \end{array} $$

$\sum\limits_\mathrm{min}^\mathrm{max}\quad\sum_\mathrm{min}^\mathrm{max}\quad$ $\displaystyle\sum_\mathrm{min}^\mathrm{max}$