DG steady advection-diffusion in an interval

Donea, J. & Huerta, A. (2003). Finite Element Methods for Flow Problems. \(\S 2.6.2\)

\[\begin{split} \mathbb{S} \begin{cases} \Omega = [0, 1] & \text{unit interval} \\ u_{\text{D}}(x=0,)=0 \\ u_{\text{D}}(x=1)=1 \\ \textbf{a}=a\,\textbf{e}_x \\ \mathsf{D}=D\mathsf{I} \\ R=0 \\ J=0 \\ u_{\text{e}}(x)=... \\ Pe= \frac{a}{2DN_x} & \text{local Peclet number} \\ \end{cases} \end{split}\]

import numpy as np
from ufl.core.expr import Expr
from ufl import SpatialCoordinate, as_vector, cos, sqrt
from lucifex.mesh import rectangle_mesh
from lucifex.fem import Function, Constant
from lucifex.solver import bvp, BoundaryConditions, BoundaryValueProblem
from lucifex.viz import plot_colormap, plot_line, save_figure
from lucifex.utils import cross_section
from lucifex.pde.advection_diffusion import (
    steady_advection_diffusion, 
    dg_steady_advection_diffusion
)