Avoid repeated assembly of multilinear forms

Question

Dear all,

I am trying to apply the ideas of the "Avoid Assembly" paragraph from the "Time-Dependent problems" tutorial to speed up a slightly more complicated problem.

One of the form I would like to pre-assemble looks like the following:

mesh = RectangleMesh (0, 0, 1, 1, 10, 10)
V = VectorFunctionSpace(mesh, "CG", 1)
g = Function(V)
u = TrialFunction(V)
v = TestFunction(V)

a = inner(g, u)*inner(g, v)*dx
A = assemble(a)

The values of the function g is changing at each iteration of the computation. So the assembly is done from zero at every iteration. Is there a way to assemble a matrix (or several) that would be multiplied by g to get A so that the repeated assembly of A can be avoided ?

Answer 1

Hi,

Unfortunately, there's no immediate way to do what you propose if your form is correct. Basically, the avoiding assemble trick is used to compute the assembling of linear forms quickly in terms of a matrix-vector product. You can do

A = assemble(inner(u, v)*dx)

and then get

G = assemble(inner(g, v)*dx)

with the short and fast

G = A*g.vector()

But the nonlinear in g and bilinear form you suggest needs to be reassembled each timestep.

Comments

Thank you for your answer. This is so bad !! Could there be a solution in the direction of mixed spaces ?

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Sorry, nonlinear equations are tough and I don't know of any solution in the direction of mixed spaces.

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Ok. Let me note that the equation obtained by a = inner(g, u)*inner(g, v)*dx is linear, since g is known. It is the assembly that would be multilinear, so the matrix A produced would actually be a tensor instead of a matrix...

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Ok, I assumed your g was a function of the solution and thus a regular nonlinear problem that had been linearized. Unfortunately you cannot store tensors of rank higher than 2 with FEniCS, and I guess that would be required for the fast assembling trick of multilinear forms.