Hi Daniel,

Nice to see you on here.

The "F == 0" syntax is usually reserved for nonlinear problems, though it can also be used for linear problems (which I think is what you have?)

However, when using this syntax, F should only contain Functions and TestFunctions, not TrialFunctions. The Jacobian (which corresponds to the LHS if F was linear) is then generated automatically using automatic differentiation of F with respect to an appropriate variable. E.g., in the Burgers example, we have

F = (inner((u - u_)/timestep, v)
+ inner(dot(u,nabla_grad(u)), v) + nu*inner(grad(u), grad(v)))*dx

Note that u was declared as a Function. Then a few lines later,

while (t <= end):
solve(F == 0, u)

The second argument to the solve call, u, is the variable to solve for (and where the result gets put), and the AD of F is done with respect to this variable.

However, when using the "a == L" syntax for linear problems, a must contain TrialFunctions and TestFunctions (and, optionally, Functions), while L must contain TestFunctions (and usually Functions), and so your other variant works.

Best,
Andrew

On 25 April 2016 at 08:59, Daniel Ruprecht <D.Ruprecht@leeds.ac.uk> wrote:

Dear Firedrake team,

I just began to use firedrake and tried to implement a simple heat
equation to get started. I encountered some strange (strange for me that
is) behaviour when using zero as a right hand side in a linear problem.

Following the nonlinear Burgers' equation example on the website, I
tried to write the diffusion equation u_t = nu*Delta(u) with backward
Euler as

a = (inner((u - u_)/timestep, v) + nu*inner(grad(u), grad(v))) * dx

(removing the nonlinear terms essentially) but when solving

solve(a == 0, temp)

this generates a ValueError: Provided residual is not a linear form.

If instead I put the u_ term on the right hand side

a = (inner(u, v) + timestep * nu * inner(grad(u), grad(v))) * dx
L = inner(u_, v) * dx
solve(a == L, temp)

everything works fine. Is this working as intended? I could not find a
comment about this on the website, I was confused because the nonlinear
example seemed to suggest the first variant should be the way to do it.

I attached the script I used to provide a minimum example. This is with
firedrake/2016-03, i.e. the version from March this year and Python 2.7.9.

Cheers, Daniel

--
Dr. Daniel Ruprecht
University Academic Fellow
School of Mechanical Engineering, Room 4.50
University of Leeds
Leeds LS2 9JT, UK
Email: d.ruprecht@leeds.ac.uk
Phone: +44 (0)113 343 22 01