A FEniCS Tutorial - FEniCS Project

The resulting program has the name d6_p2D.py and computes error norms in
various ways. Running this program for elements of first degree and ω = 1
yields the output
h=1.25E-01 E=3.25E-02 r=1.83
h=6.25E-02 E=8.37E-03 r=1.96
h=3.12E-02 E=2.11E-03 r=1.99
h=1.56E-02 E=5.29E-04 r=2.00
h=7.81E-03 E=1.32E-04 r=2.00
h=3.79E-03 E=3.11E-05 r=2.00
That is, we approach the expected second-order convergence of linear Lagrange
elements as the meshes become sufficiently fine.
Running the program for second-degree elements results in the expected
value r = 3,
h=1.25E-01 E=5.66E-04 r=3.09
h=6.25E-02 E=6.93E-05 r=3.03
h=3.12E-02 E=8.62E-06 r=3.01
h=1.56E-02 E=1.08E-06 r=3.00
h=7.81E-03 E=1.34E-07 r=3.00
h=3.79E-03 E=1.53E-08 r=3.00
However, using (u - u_e)**2 for the error computation, which implies interpolating
u_e onto the same space as u, results in r = 4 (!). This is an example
where it is important to interpolate u_e to a higher-order space (polynomials
of degree 3 are sufficient here) to avoid computing a too optimistic convergence
rate.
Running the program for third-degree elements results in the expected value
r = 4:
h= 1.25E-01 r=4.09
h= 6.25E-02 r=4.03
h= 3.12E-02 r=4.01
h= 1.56E-02 r=4.00
h= 7.81E-03 r=4.00
Checking convergence rates is the next best method for verifying PDE codes
(the best being exact recovery of a solution as in Section 1.6 and many other
places in this tutorial).
Flux Functionals. To compute flux integrals like (29) we need to define the
n vector, referred to as facet normal in FEniCS. If Γ is the complete boundary
we can perform the flux computation by
n = FacetNormal(mesh)
flux = -p*dot(nabla_grad(u), n)*ds
total_flux = assemble(flux)
Although nabla_grad(u) and grad(u) are interchangeable in the above expression
when u is a scalar function, we have chosen to write nabla_grad(u)
because this is the right expression if we generalize the underlying equation
to a vector Laplace/Poisson PDE. With grad(u) we must in that case write
dot(n, grad(u)).
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