A FEniCS Tutorial - FEniCS Project

The grad_u_x and grad_u_y variables behave as Function objects. In particular,
we can extract the underlying arrays of nodal values by
grad_u_x_array = grad_u_x.vector().array()
grad_u_y_array = grad_u_y.vector().array()
The degrees of freedom of the grad_u vector field can also be reached by
grad_u_array = grad_u.vector().array()
but this is a flat numpy array where the degrees of freedom for the x component
of the gradient is stored in the first part, then the degrees of freedom of the y
component, and so on.
The program d5_p2D.py extends the code d5_p2D.py from Section 1.6 with
computationsandvisualizationsofthegradient. Examiningthearraysgrad_u_x_array
andgrad_u_y_array, orlookingattheplotsofgrad_u_xandgrad_u_y, quickly
revealsthatthecomputedgrad_ufielddoesnotequaltheexactgradient(2x,4y)
in this particular test problem where u = 1+x 2 +2y 2 . There are inaccuracies at
the boundaries, arising from the approximation problem for w. Increasing the
mesh resolution shows, however, that the components of the gradient vary linearly
as 2x and 4y in the interior of the mesh (i.e., as soon as we are one element
away from the boundary). See Section 1.8 for illustrations of this phenomenon.
Projecting some function onto some space is a very common operation in
finite element programs. The manual steps in this process have therefore been
collected in a utility function project(q, W), which returns the projection of
some Function or Expression object named q onto the FunctionSpace or
VectorFunctionSpace named W. Specifically, the previous code for projecting
each component of grad(u) onto the same space that we use for u, can now be
done by a one-line call
grad_u = project(grad(u), VectorFunctionSpace(mesh, ’Lagrange’, 1))
The applications of projection are many, including turning discontinuous gradient
fields into continuous ones, comparing higher- and lower-order function
approximations, and transforming a higher-order finite element solution down
to a piecewise linear field, which is required by many visualization packages.
1.10 A Variable-Coefficient Poisson Problem
Suppose we have a variable coefficient p(x,y) in the Laplace operator, as in the
boundary-value problem
−∇·[p(x,y)∇u(x,y)] = f(x,y) in Ω,
u(x,y) = u 0 (x,y) on ∂Ω.
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We shall quickly demonstrate that this simple extension of our model problem
only requires an equally simple extension of the FEniCS program.
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