One-dimensional stationary convection–diffusion model and uniform P1 space

Fix an integer $N\ge 1$, put $h=1\mathrm{/}N$, and let $x_j=jh$ for $j=0,\dots ,N$. The cells are $K_j=[x_{j-1},x_j]$. For $g_0,g_1\in \mathbb{R}$, let $S_h(g_0,g_1)$ be the finite-dimensional affine space of continuous functions on $[0,1]$ that are affine on each cell and have endpoint values $g_0,g_1$. Write $V_h=S_h(0,0)$ for the homogeneous test and trial vector space; explicitly, $S_h(g_0,g_1)=\mathrm{\ell }+V_h$ with $\mathrm{\ell }(x)=(1-x)g_0+x{g}_1$. “P1” means polynomial degree at most one on each interval.

Let $ϵ>0$, $b\in \mathbb{R}\setminus \{0\}$, and $f\in C([0,1])$. The stationary boundary-value problem is to find $u\in C^2([0,1])$ such that $-ϵu^{\mathrm{\prime }\mathrm{\prime }}+b{u}^{\mathrm{\prime }}=f$ on $(0,1)$, with prescribed endpoint values $u(0)=g_0$ and $u(1)=g_1$. Its dimensionless cell Péclet number is $\mathrm{P}\mathrm{e}=\mathrm{\mid }b\mathrm{\mid }h\mathrm{/}(2ϵ)>0$. The stability theorem on this page uses the homogeneous branch $g_0=g_1=0$; the general endpoint notation also permits the boundary-layer example.