Let
- $a,b\in\mathbb R$ with $a<b$
- $\Lambda:=(a,b)$
- $n\in\mathbb N$
- $h:=(b-a)/n$
- $z_k:=a+kh$ for $k\in\left\{0,\ldots,n\right\}$
- $e_k:=[z_{k-1},z_k]$
- $r\in\mathbb N$
Moreover, let $$\tilde V:=\left\{v\in C^0(\overline\Lambda):\left.v\right|_{e_k}\in\mathbb P_r(e_k)\text{ for all }k\in\left\{1,\ldots,n\right\}\text{ and }\left.v\right|_{\partial\Lambda}=0\right\}$$ where $\mathbb P_r(e_k)$ denotes the set of real-valued polynomials on $e_k$ of degree at most $r$.
Now, let $J\in\mathbb N$ and $\phi_1,\ldots,\phi_J:\overline\Lambda\to\mathbb R$ with $$\left.\phi_j\right|_{e_k}\in\mathbb P_r(e_k)\text{ and }\phi_j(x_i)=\delta_{ij}\;\;\;\text{for all }i,j\in\left\{1,\ldots,J\right\}\text{ and }k\in\left\{1,\ldots,n\right\}\tag 1$$ (where $\delta_{ij}$ denotes the Kronecker delta) for some $x_1,\ldots,x_J\in\overline\Lambda$.
In An Introduction to Computational Stochastic PDEs the authors claim on page 50 that $$\tilde V=\operatorname{span}\left\{\phi_1,\ldots,\phi_J\right\}\tag 2\;.$$
Is this really true?
Let's take a look: Let $v\in\tilde V$ and $$w(x):=\sum_{j=1}^Jw_j\phi_j\;\;\;\text{for }x\in\overline\Lambda$$ for some $w_1,\ldots,w_J\in\mathbb R$ to be determined such that $v=w$.
- Let $$p:=v-w$$
- We obtain $$p(x_i)=v(x_i)-w_i\;\;\;\text{for all }i\in\left\{1,\ldots,J\right\}\tag 3$$
- So, we may choose $$w_i:=v(x_i)\;\;\;\text{for }i\in\left\{1,\ldots,J\right\}$$
- Let $k\in\left\{1,\ldots,n\right\}$ $\Rightarrow$ $$\left.p\right|_{e_k}\in\mathbb P_r(e_k)$$
Now, I think that we need the additional assumption that $$\left|\left\{x_1,\ldots,x_J\right\}\cap e_k\right|=r+1\;,\tag 4$$ since in that case $p$ has at least $r+1$ distinct roots by $(3)$, which is impossible for a nontrivial polynomial of degree at most $r$. So, $\left.p\right|_{e_k}=0$ and hence $$\left.v\right|_{e_k}=\left.w\right|_{e_k}\;.\tag 5$$
Question 1: Am I right? Do we really need $(4)$ or can we show $(5)$ in general (as claimed by the authors)?
Question 2: The authors state that $x_1,\ldots,x_J$ have to be chosen such that $\phi_1,\ldots,\phi_J$ are in $C^0(\overline\Lambda)$. I don't understand what they mean. By definition, each $\phi_j$ is a polynomial on each $e_k$ and hence piecewise continuous. Now, $\overline\Lambda=\bigcup_{k=1}^ne_k$ and since each $e_k$ is closed there is no problem at the points $z_k$. In other words, the polynomials on $e_k$ and $e_{k+1}$ agree at $z_k$. Since this is true for all $k$, we should already be able to conclude the continuity on $\overline\Lambda$. What am I missing?
Question 3: What can we say about $\left.\phi_j\right|_{\partial\Lambda}$? Do we know that $\left.\phi_j\right|_{\partial\Lambda}=0$ or do we need an additional assumption?
(comment) Of course, dimension of $V$ depends on $r$ and therefore, $J$ must be connected with $r$ so that dimensions agree. The problem with continuity can arise if you want to define a $\varphi_j$ on the entire interval. You have a polynomial to the left of $z_k$ and another one to the right. It doesn't follow that that they have the same value at $z_k$ if $x_j$ are all different from $z_k$. Just consider the case of two intervals abd $r=1$ with $x_j$'s inside the intervals. You will have two linear functions which might not coincide at the boundary point between the intervals. In case of general $x_j$ boundary conditions should be imposed explicitly since in general it will not follow from anywhere else.