For 1D derivative we have \begin{equation} F(x) = \frac{\partial f(x)}{\partial x} \end{equation} \begin{equation} f(x)=\sum_{i}f_ie_i(x) \end{equation} \begin{equation} F(x)=\sum_{i}F_ie_i(x) \end{equation} where $e_i(x)$ are the FEM basis functions. We can then apply the Galerkin procedure to the derivative,and then get the matrix form \begin{equation} A\widehat{F}=B\widehat{f} \end{equation} \begin{equation} \widehat{F}=A^{-1}B\widehat{f} \end{equation} $\widehat{F}$and$\widehat{f}$are the vector form of $F_i$and $f_i$.
For the derivative of the product of two functions $f(x)$and $g(x)$,applying product rule we get \begin{equation} \frac{\partial (f(x).g(x))}{\partial x}=f(x). \frac{\partial g(x)}{\partial x}+g(x). \frac{\partial f(x)}{\partial x} \end{equation}
Now in FEM I want the derivative matrix $C=A^{-1}B$ also to fulfill the similar constraint \begin{equation} C \widehat{fg} = [f]_{dig}.C\widehat{g} + [g]_{dig}.C\widehat{f} \end{equation} where $[f]_{dig}$ means diagonal matrix with the values $f_i$ in it.
Does any body know what kinds of basis function $e_i(x)$ I can chose?
