One-dimensional residual SUPG form with the exact cell parameter

Use the uniform mesh, coefficients, and spaces of One-dimensional stationary convection–diffusion model and uniform P1 space. On every cell set $\tau =\frac{h}{2\mathrm{\mid }b\mathrm{\mid }}(\coth (\mathrm{P}\mathrm{e})-\frac{1}{\mathrm{P}\mathrm{e}}),\coth s:=\frac{\cosh s}{\sinh s}\mathrm{.}$ For $w\in S_h(g_0,g_1)+C^2([0,1])$ and $v_h\in V_h$, define $a_h(w,v_h):=ϵ{\int }_0^1{w}^{\mathrm{\prime }}{v}_h^{\mathrm{\prime }}dx+b{\int }_0^1{w}^{\mathrm{\prime }}{v}_{h}dx+{\sum }_{K\in {\mathcal{T}}_h}\tau {\int }_K(-ϵw^{\mathrm{\prime }\mathrm{\prime }}+b{w}^{\mathrm{\prime }})(b{v}_h^{\mathrm{\prime }})dx,$ $F_h(v_h):={\int }_0^{1}f{v}_{h}dx+{\sum }_{K\in {\mathcal{T}}_h}\tau {\int }_{K}f(b{v}_h^{\mathrm{\prime }})dx\mathrm{.}$ Second derivatives are taken classically on each cell. The strong equation residual is $R_K(w):=-ϵw^{\mathrm{\prime }\mathrm{\prime }}+b{w}^{\mathrm{\prime }}-f$. For P1 trial functions, $w_h^{\mathrm{\prime }\mathrm{\prime }}=0$ on each cell, so the cell term is $\tau {\int }_K(b{w}_h^{\mathrm{\prime }})(b{v}_h^{\mathrm{\prime }})$. On the smooth consistency extension it retains $-ϵw^{\mathrm{\prime }\mathrm{\prime }}$; thus the full residual pairing is $\tau {\int }_K(R_K(w)+f)(b{v}_h^{\mathrm{\prime }})$, while the load contains the matching $f$ term. The SUPG equation is $a_h(w_h,v_h)=F_h(v_h)$ and adds the same residual-weighted streamline test perturbation to the differential equation and its load.