Using manufactured solutions to verify the accuracy of a mixed function space problem

Question

In a time-dependent mixed function space problem, I wanna to compare the calculated solution with analytical solution, but haven't got the right way.

I know for a one unknown problem, we use a script like this one:

u_exact = interpolate(g, V)
maxdiff = numpy.abs(u_exact.vector().array() - u.vector().array()).max()
print 'Max error, t=%.2f: %-10.3f' % (t, maxdiff)

What if it becomes a mixed function space? How to compare each of them? How to interpolate the analytical solution?

In my problem, the mixed function space is:

Q = FunctionSpace(mesh, "CG", 2)
S = FunctionSpace(mesh, "CG", 1)
W = Q*S

and the function w is split into two parts as

 w     = Function(W)
(p,s)  = split(w)   # current solution

and the while loop for time stepping is like:

while t <= T:
    print t
    solve(F == 0, w, bc)
    w0.vector()[:] = w.vector()
    t += float(dt)

Answer 1

See here for an answer on how to interpolate in mixed function spaces.

For mixed FEM for Navier-Stokes equations I have an example in this preprint (Section 4).

Btw., I prefer python's scipy module to construct analytical solutions and corresponding source terms, while others stick to UFL, see this post.

Comments

Thanks, Jan. It solves my problem.

Answer 2

You can use SymPy to construct a right-hand side matching any solution that you choose, and then use Dolfin's Expressions to translate the SymPy solution into something that you can use in Dolfin.

See Maelstrom's testing suite for an example.