$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A note on our notation and the different meanings of \( u \) (2)
Or we may approximate \( u \): \( u(\x) = \sum_jc_j\baspsi_j(\x) \) and
let this spatially discrete \( u \) enter the variational form,
$$
\int_\Omega (uv + \Delta t\,\dfc(u)\nabla u\cdot\nabla v
- \Delta t f(u)v - u^{(1)} v)\dx,\quad\forall v\in V
$$
Picard iteration: \( u(\x) \) solves the approximate
variational form
$$
\int_\Omega (uv + \Delta t\,\dfc(u^{-})\nabla u\cdot\nabla v
- \Delta t f(u^{-})v - u^{(1)} v)\dx
$$
Could introduce
- \( \uex(\x,t) \) for the exact solution of the PDE problem
- \( \uex(\x)^n \) for the exact solution after time discretization
- \( u^n(\x) \) for the spatially discrete solution \( \sum_jc_j\baspsi_j \)
- \( u^{n,k} \) for approximation in Picard/Newton iteration no \( k \)
to \( u^n(\x) \)
