Problem when using slate Tensor

[marazzaf]

Hi everyone,

I am trying to learn how to use Slate in order to do static condensation.

But when I try to run a very simple example, I run into issues. Here is a minimal working example.

from firedrake import *
from firedrake.slate import Tensor, AssembledVector, Solve, Block
import sys

#Mesh
N = 25
mesh = UnitSquareMesh(N, N, diagonal='crossed')

#Spaces
V = FunctionSpace(mesh, "DG", 1)           # Scalar u
W = VectorFunctionSpace(mesh, "DG", 1)     # Flux sigma

#Mixed function space
Z = V * W
PETSc.Sys.Print('Nb dof: %i' % Z.dim())

#Trial and test functions
u,sigma = TrialFunctions(Z)
v,tau = TestFunctions(Z)

#Geometric functions
n = FacetNormal(mesh)
h = CellDiameter(mesh)

#RHS
f = Function(V)
x, y = SpatialCoordinate(mesh)
f.interpolate(2 * pi**2 * sin(pi * x) * sin(pi * y))  # exact solution: sin(pi x) sin(pi y)
L = f * v * dx

#LHS
a_interior = dot(sigma, tau)*dx + dot(grad(u), tau)*dx - dot(sigma, grad(v))*dx

#Interior penalty terms
alpha = Constant(10)
a_interior += alpha / avg(h) * jump(u)*jump(v)*dS - inner(dot(avg(tau), n('+')), jump(u))*dS + inner(dot(avg(sigma), n('+')), jump(v))*dS

#Boundary terms (homogeneous Dirichlet)
a_boundary = alpha / h * u * v * ds - dot(tau, n) * u * ds + dot(sigma, n) * v * ds

#Total bilinear form and RHS
a = a_interior + a_boundary

#Slate
A = Tensor(a)
b = Tensor(L)

#Solving
z = Function(Z, name="z")
solve(assemble(A), z.vector(), assemble(b))
#solve(assemble(a), z.vector(), assemble(L))

#Store result
from firedrake.output import VTKFile
file = VTKFile('sol.pvd')
file.write(z.sub(0))

If you comment the solve and use the "regular" solve without creating the Tensor objects before you get the expected solution. But if you assemble the Tensor objects and then solve, the computed solution is very wrong. I don't understand why.

Thank you for your help!

[connorjward]

@pbrubeck any insights?

[pbrubeck]

As far as I understand, Slate is designed to work on hybridised DG mixed formulations of Poisson (as discussed in the paper and this demo). The IPDG formulation in your example is not hybridasable, which essentially means that you cannot include cross terms like u("+")*v("-") arising from jump(u)*jump(v), only u("+")*v("+") and u("-")*v("-") are supported by Slate. Unfortunately Slate does not run checks on this and simply gives the wrong answer.

Apart from being able to deal with hybridised DG, slate can be used to statically-condense continuous elements, see this test.

[marazzaf]

@pbrubeck: Thank you for your explanation. I did not know about these limitations. Will these limitations ever be lifted? It would be nice to be able to use Slate operations on any kind of formulation.

Since I cannot use Slate for this problem, should I use the solver options like in this example?

[pbrubeck]

@pbrubeck: Thank you for your explanation. I did not know about these limitations. Will these limitations ever be lifted? It would be nice to be able to use Slate operations on any kind of formulation.

Unfortunately there are no plans to further develop Slate to support non-hybridasable formulations, as far as I can tell.

Since I cannot use Slate for this problem, should I use the solver options like in this example?

That looks like a good place to start

[marazzaf]

Guess that's where I'll start then.
Thanks y'all!

[dham]

Hi all. Slate enables local linear algebra on cell integrals. This only works when the cell integrals are decoupled from each other. Surface terms with mixed restrictions have the effect of coupling adjacent cells, which is incompatible with this abstraction.

It is possible to think of extending Slate to facet integrals, which would enable a certain class of problems where there are mixed restrictions (though still not all of them, because care would need to be taken to avoid coupling adjacent facets). There are, as @pbrubeck says, no current plans to do this.

[marazzaf]

Thank you for your answer @dham. I think that what you propose would be a welcomed extension to Slate but I understand that it is not in the books for now.