How to assemble multilinear form

Question

I'm trying to assemble a multilinear form (for no purpose other than understanding better what's happening). From the FEniCS book, I understand that a multilinear form of arity-2 should result in a rank 2 tensor after assembly. So I tried to assemble something trivial:

# (define v1 and v2)
A = assemble(v1 * v2 * dx)

I was not quite sure how assembly depends on v1 and v2 being test functions or trial functions (at this stage, there didn't seem to be any difference), so I opted for test functions.

However, none of the following definitions worked at all:

# First try
V = FunctionSpace(mesh, 'CG', 1)
(v1, v2) = (TestFunction(V), TestFunction(V))
A = assemble(v1 * v2 * dx) # "Invalid expression"

# Second try
V = FunctionSpace(mesh, 'CG', 1)
(v1, v2) = TestFunction(V*V)
A = assemble(v1 * v2 * dx) # "Unable to extract all indices."

# Third try
V = VectorFunctionSpace(mesh, 'CG', 1)
v = TestFunction(V)
(v1, v2) = (v[0], v[1])
A = assemble(v1 * v2 * dx) # "Unable to extract all indices."

Could someone explain to me what's wrong with each of these, and how to do it right?

Answer 1

Your mistakes in comments below

# First try
V = FunctionSpace(mesh, 'CG', 1)
(v1, v2) = (TestFunction(V), TestFunction(V))
A = assemble(v1 * v2 * dx) # "Invalid expression"
# Is nonlinear

# Second try
V = FunctionSpace(mesh, 'CG', 1)
(v1, v2) = TestFunction(V*V)
A = assemble(v1 * v2 * dx) # "Unable to extract all indices."
# What is a multiplication of mixed fuctions?
# Moreover, is nonlinear

# Third try
V = VectorFunctionSpace(mesh, 'CG', 1)
v = TestFunction(V)
(v1, v2) = (v[0], v[1])
A = assemble(v1 * v2 * dx) # "Unable to extract all indices."
# Is nonlinear

Examples of correct bilinear (rank 2 multilinear forms):

# Arbitrary scalar space
V = FunctionSpace(...)

# Let's assemble L^2 inner product
u, v = TrialFunction(V), TestFunction(V)
A = assemble(u*v*dx)

# Or assemble any other form which is linear in both u, v,
# for example H_0^1 inner product
A = assemble(inner(grad(u), grad(v))*dx)

Comments

How can u*v*dx be both linear and non-linear, depending on u being a test or trial function? As far as I know, a multilinear form distinguishes only between coefficient functions and argument functions, but it should not matter at all if an argument represents a test function or a trial function in some variational problem.

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In FEniCS it holds:

• trial function is first argument
• test function is last argument
• multi-linear form has to be linear in each of its arguments
• multi-linear form can depend non-linearly on coefficients
• DOLFIN can assemble forms of rank
  
  
  • 0 (no arguments),
  • 1 (has (customarily) test function),
  • 2 (has trial and test function)