A FEniCS Tutorial - FEniCS Project

x, flux_x_exact, ’b-’,
legend=(’numerical (projected) flux’, ’exact flux’),
title=’Flux in x-direction (at y=%g)’ % y_fixed,
savefig=’flux.eps’)
AsseenfromFigure1.12b, thenumericalfluxisaccurateexceptintheboundary
elements.
Thevisualizationconstructionsshownaboveandusedtogeneratethefigures
are found in the program vcp2D.py in the stationary/poisson directory.
It should be easy with the information above to transform a finite element
field over a uniform rectangular or box-shaped mesh to the corresponding
BoxField object and perform MATLAB-style visualizations of the whole field
or the field over planes or along lines through the domain. By the transformation
to a regular grid we have some more flexibility than what Viper offers.
However, we remark that comprehensive tools like VisIt, MayaVi2, or ParaView
also have the possibility for plotting fields along lines and extracting planes in
3D geometries, though usually with less degree of control compared to Gnuplot,
MATLAB, and Matplotlib. For example, in investigations of numerical
accuracy or numerical artifacts one is often interested in studying curve plots
where only the nodal values sampled. This is straightforward with a structured
mesh data structure, but more difficult in visualization packages utilizing unstructured
grids, as hitting exactly then nodes when sampling a function along
a line through the grid might be non-trivial.
1.13 Combining Dirichlet and Neumann Conditions
Let us make a slight extension of our two-dimensional Poisson problem from
Section 1.1 and add a Neumann boundary condition. The domain is still the
unit square, but now we set the Dirichlet condition u = u 0 at the left and right
sides, x = 0 and x = 1, while the Neumann condition
− ∂u
∂n = g
is applied to the remaining sides y = 0 and y = 1. The Neumann condition is
alsoknownasanatural boundary condition (incontrasttoanessentialboundary
condition).
Let Γ D and Γ N denote the parts of ∂Ω where the Dirichlet and Neumann
conditions apply, respectively. The complete boundary-value problem can be
written as
−∇ 2 u = f in Ω, (30)
u = u 0 on Γ D , (31)
− ∂u
∂n = g on Γ N . (32)
Again we choose u = 1+x 2 +2y 2 as the exact solution and adjust f, g, and u 0
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