Multidimensional P1 SUPG residual form and parameter scaling

Use the model and notation of Stationary multidimensional convection–diffusion–reaction P1 model. Define $s_b(K)=h_K\mathrm{/}\mathrm{\parallel }b{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$ when that norm is positive and $s_b(K)=+\mathrm{\infty }$ otherwise; define $s_c(K)=\omega \mathrm{/}\mathrm{\parallel }c{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}^2$ when that norm is positive and $s_c(K)=+\mathrm{\infty }$ otherwise. For each cell set ${\delta }_K:=\theta \min \{\frac{h_K^2}{ϵ},s_b(K),s_c(K)\},0<\theta \le \frac{1}{4}\mathrm{.}$ The minimum is finite and positive because its first entry is finite and positive. This scaling gives ${\delta }_K\mathrm{\parallel }c{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}^2\le \omega \mathrm{/}4$.

For a trial function $w$ that is the sum of a cellwise smooth function and a member of $V_h$, and $v_h\in V_h$, define the element residual $R_K(w):=-ϵ{\mathrm{\Delta }}_{K}w+b\cdot \mathrm{\nabla }w+cw$, and $a_h(w,v_h):=ϵ(\mathrm{\nabla }w,\mathrm{\nabla }{v}_h{)}_{\mathrm{\Omega }}+(b\cdot \mathrm{\nabla }w+cw,v_h{)}_{\mathrm{\Omega }}+{\sum }_{K\in {\mathcal{T}}_h}{\delta }_K(R_K(w),b\cdot \mathrm{\nabla }{v}_h{)}_K,$ $F_h(v_h):=(f,v_h{)}_{\mathrm{\Omega }}+{\sum }_{K\in {\mathcal{T}}_h}{\delta }_K(f,b\cdot \mathrm{\nabla }{v}_h{)}_K\mathrm{.}$ Here $(g_1,g_2{)}_D:={\int }_D{g}_1{g}_{2}dx$ and ${\mathrm{\Delta }}_K$ is the classical Laplacian restricted to the interior of $K$. For $w_h\in V_h$, ${\mathrm{\Delta }}_K{w}_h=0$ because $w_h$ is affine on $K$. No stability claim is part of this definition.