Say we are working in the grid $[0,1]\times[0,1]$ in the $xy-$plane. The 2D FEM basis function definition I have says $\psi_{ab}(x,y) = \phi_{a}(x)\phi_{b}(y)$ for $1\leq a,b \leq N-1$ with
\begin{equation*} {\phi _a (x) =\left\{ \begin{array}{ll} \frac{x-x_{a-1}}{h} & x\in [(a-1)h,ah] \\ \frac{x_{a+1}-x}{h} & x \in [ah,(a+1)h] \\ 0 & \mbox{else} \end{array} \right.} \mbox{ }\mbox{ } \mbox{ }\mbox{ }\mbox{ }\mbox{ for } a = 1,...,N-1 \end{equation*}
- $N = $ number of intervals we cut the full interval $[0,1]$ into (can change for x and y intervals)
- $h = 1/N$
- $x_{j} = jh$
I used the 2D FEM Basis functions, $\psi (x,y)$, described above to make a basis representation for a scalar function u(x,y). I now want to create a basis to represent a vector function v(x,y)$\in \mathbb{R}^2$ for a new problem . My current thinking is that if I have some vector function
$ \begin{equation*} v (x,y) =\left( \begin{array}{ll} v^1 (x,y) \\ v^2(x,y) \end{array} \right) \end{equation*} $
I think I can represent this function as $ v = \sum _ {a,b = 1} ^ {N-1} V_{a,b} \hat{\psi}_{ab}(x,y)$ where $V_{a,b}$ are coefficients and
$ \begin{equation*} \hat{\psi}_{ab}(x,y) =\left( \begin{array}{ll} \phi_{a}^1(x)\phi_{b}^1(y) \\ \phi_{a}^2(x)\phi_{b}^2(y) \end{array} \right) \end{equation*} $
My thinking is if we know that $\psi(x,y)$ is a basis (since I was given the first definition I wrote), then we know that $c_{11} \psi_{11} + c_{12} \psi_{12}+...+c_{N-1,N-1} \psi_{N-1,N-1} = 0 $
if and only if
$ c_{11} = c_{12} = c_{13}=...=c_{N-1,N-1}=0$.
So I would think that if you took a linear combination of the new basis I wrote for the vector valued basis functions we would get $c_{11} \hat{\psi}_{11} + c_{12} \hat{\psi}_{12}+...+c_{N-1,N-1} \hat{\psi}_{N-1,N-1} = \vec{0} $ if and only if
$ c_{11} = c_{12} = c_{13}=...=c_{N-1,N-1}= \left(\begin{array}{ll} 0 \\ 0 \end{array}\right) $
Which I think shows linear independence.
- Is what I outlined the correct FEM basis to represent a function in $\mathbb{R}^2$?
- How would I even interpret the coefficients $V_{ab}$ here? Previously, in the case when I used the $\psi(x,y)$ basis functions $V_{ab}$ was a matrix. Now we get a coefficient that is a 2D vector for each combination of a and b (instead of a single number). How can I visualize this? Do I just input a vector into a matrix each time to track the coefficients?
Sorry if these are silly questions I'm struggling to fully visualize this and can't seem to find anything very helpful online. Thank you for any input.