Stationary multidimensional convection–diffusion–reaction P1 model

Fix $d\ge 2$ and a bounded connected polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$. Let $\{{\mathcal{T}}_h\}$ be a shape-regular family of finite matching affine simplicial meshes in the sense of Matching affine simplicial meshes and shape regularity: one finite constant $\sigma$ satisfies $h_K\mathrm{/}{\rho }_K\le \sigma$ for every cell of every mesh. Put $h={\max }_{K\in {\mathcal{T}}_h}{h}_K$. Let $V_h$ be the space of continuous functions on $\bar{\mathrm{\Omega }}$ that are affine on every cell and vanish on $\mathrm{\partial }\mathrm{\Omega }$.

Let $ϵ>0$, $b\in C^1(\bar{\mathrm{\Omega }};{\mathbb{R}}^d)$, $c\in C(\bar{\mathrm{\Omega }})$, and $f\in C(\bar{\mathrm{\Omega }})$. Define $B:=\mathrm{\parallel }b{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}$ and $C:=\mathrm{\parallel }c{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}$. Assume a fixed $\omega >0$ satisfies $c-\frac{1}{2}\mathrm{\nabla }\cdot b\ge \omega$ on $\mathrm{\Omega }$. The boundary condition for this branch is $u=0$ on the whole boundary, including the inflow part $\{x\in \mathrm{\partial }\mathrm{\Omega }:b(x)\cdot n(x)<0\}$. A classical exact solution is a function $u\in C^2(\bar{\mathrm{\Omega }})$ satisfying $-ϵ\mathrm{\Delta }u+b\cdot \mathrm{\nabla }u+cu=f$ in $\mathrm{\Omega }$ and this boundary condition.