human
3. Affine decomposition
For this problem the affine decomposition is straightforward:
$$a(u,v;\boldsymbol{\mu})=\underbrace{\frac{1}{10\,^{\boldsymbol{\mu}}}}{\Theta^{a}_0(\boldsymbol{\mu})}\underbrace{\int{\Omega}\nabla u \cdot \nabla v \ d\boldsymbol{x}}{a_0(u,v)} \ + \  \underbrace{1}{\Theta^{a}1(\boldsymbol{\mu})}\underbrace{\int{\Omega}\left[(1,1)\cdot\nabla u\right]v \ d\boldsymbol{x}}{a_1(u,v)},$$
$$f(v.
\boldsymbol{\mu}) = \underbrace{1}{\Theta^{f}0(\boldsymbol{\mu})} \underbrace{\int{\Omega}v \ d\boldsymbol{x}}_{f_0(v)}.$$
Adding the following forms, we obtaing the affine decomposition for the stabilized approach:
$$s(u,v;\boldsymbol{\mu}) = \sum_{K\in\mathcal{T}h}\underbrace{\frac{\delta_K}{10\,^{\boldsymbol{\mu}}}}{\Theta^{s}0(\boldsymbol{\mu})}\underbrace{\int_K 
    \Delta u\left(\frac{h_K}{\sqrt{2}}(1,1)\cdot\nabla v\right)\ d\boldsymbol{x}}{s_0(u,v)} \ + \
    \sum_{K\in\mathcal{T}h}\underbrace{\delta_K}{\Theta^{s}1(\boldsymbol{\mu})}\underbrace{\int_K 
    \left((1,1)\cdot\nabla u\right)\left(\frac{h_K}{\sqrt{2}}(1,1)\cdot\nabla v\right)\ d\boldsymbol{x}}{s_1(u,v)},$$
$$r(v.
\boldsymbol{\mu}) = \sum_{K\in\mathcal{T}h}\underbrace{\delta_K}{\Theta^{r}0(\boldsymbol{\mu})} \underbrace{\int_K\left(\frac{h_K}{\sqrt{2}}(1,1)\cdot\nabla v\right)\ d\boldsymbol{x}}{r_0(v)}.$$
We will implement the numerical discretization of the problem in the class
class AdvectionDominated(EllipticCoerciveProblem):
by specifying the coefficients $\Theta^{a}(\boldsymbol{\mu})$ and $\Theta^{f}_(\boldsymbol{\mu})$ in the method
def compute_theta(self, term):
and the bilinear forms $a(u, v)$ and linear forms $f_(v)$ in
def assemble_operator(self, term):

=====
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#Here is the Python code you requested.
@OnlineStabilization()
class AdvectionDominated(EllipticCoerciveProblem):

    # Default initialization of members
    def __init__(self, V, **kwargs):
        # Call the standard initialization
        EllipticCoerciveProblem.__init__(self, V, **kwargs)
        # ... and also store FEniCS data structures for assembly
        assert "subdomains" in kwargs
        assert "boundaries" in kwargs
        self.subdomains, self.boundaries = kwargs["subdomains"], kwargs["boundaries"]
        self.u = TrialFunction(V)
        self.v = TestFunction(V)
        self.dx = Measure("dx")(subdomain_data=subdomains)
        self.ds = Measure("ds")(subdomain_data=boundaries)
        # Store advection and forcing expressions
        self.beta = Constant((1.0, 1.0))
        self.f = Constant(1.0)
        # Store terms related to stabilization
        self.delta = 0.5
        self.h = CellDiameter(V.mesh())

    # Return custom problem name
    def name(self):
        return "AdvectionDominated1POD"

    # Return theta multiplicative terms of the affine expansion of the problem.
    def compute_theta(self, term):
        mu = self.mu
        if term == "a":
            theta_a0 = 10.0**(- mu[0])
            theta_a1 = 1.0
            if self.stabilized:
                delta = self.delta
                theta_a2 = - delta * 10.0**(- mu[0])
                theta_a3 = delta
            else:
                theta_a2 = 0.0
                theta_a3 = 0.0
            return (theta_a0, theta_a1, theta_a2, theta_a3)
        elif term == "f":
            theta_f0 = 1.0
            if self.stabilized:
                delta = self.delta
                theta_f1 = delta
            else:
                theta_f1 = 0.0
            return (theta_f0, theta_f1)
        else:
            raise ValueError("Invalid term for compute_theta().")

    # Return forms resulting from the discretization of the affine expansion of the problem operators.
    def assemble_operator(self, term):
        v = self.v
        dx = self.dx
        if term == "a":
...