$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Fundamental integration problem: how to deal with \( \int f(\sum_kc_k\baspsi_k)\baspsi_idx \) for unknown \( c_k \)?
- We do not know \( c_k \) in \( \int_0^L f(\sum_kc_k\baspsi_k)\baspsi_i dx \) and
\( \int_0^L \dfc(\sum_{k}c_k\baspsi_k)\baspsi_i^{\prime}\baspsi_j^{\prime}\dx \)
- Solution: numerical integration with approximations to \( c_k \),
as in \( \int_0^L f(u^{-})\baspsi_idx \)
Next: want to do symbolic integration of such terms to see the
structure of nonlinear finite element equations (to compare with
finite differences)
