Let $u$ be the unique solution of problem (1) and $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ the family defined by relation (2) for $n \in \mathbb { N } ^ { * }$. Show that there exists a constant $\tilde { C } > 0$, independent of $n$, such that $$\max _ { 0 \leq i \leq n + 1 } \left| u \left( x _ { i } \right) - u _ { i } \right| \leq \frac { \tilde { C } } { n ^ { 2 } }$$ Hint: one may introduce the vector $X = {}^{ t } \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right)$ where we set $\varepsilon _ { i } = u \left( x _ { i } \right) - u _ { i }$ and compute $A _ { n } X$.
Let $u$ be the unique solution of problem (1) and $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ the family defined by relation (2) for $n \in \mathbb { N } ^ { * }$. Show that there exists a constant $\tilde { C } > 0$, independent of $n$, such that
$$\max _ { 0 \leq i \leq n + 1 } \left| u \left( x _ { i } \right) - u _ { i } \right| \leq \frac { \tilde { C } } { n ^ { 2 } }$$
Hint: one may introduce the vector $X = {}^{ t } \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right)$ where we set $\varepsilon _ { i } = u \left( x _ { i } \right) - u _ { i }$ and compute $A _ { n } X$.