Integration of different kinds of forms

Integration of different kinds of forms#

As we have seen in the section on Integration measures, we have used dolfinx.fem.assemble_scalar() to compute scalar integrals in DOLFINx. However, in the subsection The variational form of a projection we observed that we need to compute a matrix \(A\) and a vector \(b\). How do we compute them in DOLFINx without using dolfinx.fem.petsc.LinearProblem?

In this section, we will compare assembly on a linear and a curved mesh. We start by creating these two meshes with the function from the previous section.

from benefits_of_curved_meshes import generate_mesh

resolution = 0.01
linear_mesh, _, _ = generate_mesh(resolution, 1)
curved_mesh, _, _ = generate_mesh(resolution, 3)

For this section, we will use the following finite element definition

element = ("Lagrange", 3, (2,))

and the function spaces

import dolfinx

V = dolfinx.fem.functionspace(curved_mesh, element)
V_lin = dolfinx.fem.functionspace(linear_mesh, element)

Assembling a vector#

Then, we will consider the right hand side of the variational form:

\[ \int_\Omega f \cdot v ~\mathrm{d}x\qquad \forall v \in V. \]

We make a convenience function to create this form

import ufl


def linear_form(f, v, dx):
    return ufl.inner(f, v) * dx

Assembly on the curved mesh#

We then define the right hand side equation over the curved mesh

v = ufl.TestFunction(V)
f = dolfinx.fem.Constant(curved_mesh, (0, -9.81))
dx_curved = ufl.Measure("dx", domain=curved_mesh)
L = linear_form(f, v, dx_curved)

We compile the form

L_compiled = dolfinx.fem.form(L)

We can now assemble the right hand side vector using dolfinx.fem.petsc.assemble_vector() Additionally, we compute the runtime of the assembly by using the time module. Since we want to assemble a vector, we pre-define it as a dolfinx.fem.Function from the function space of the test function, so that we can re-use it for repeated assemblies. Additionally, this takes care of some of the memory management when interfacing with PETSc.

import dolfinx.fem.petsc
from time import perf_counter

b = dolfinx.fem.Function(V)

Accumulation of values when assembling a vector

When we call assemble_vector, we will not zero out values already present in the vector. Instead, the new values will be added to the existing values. Call b.x.array[:] = 0.0 to zero out the vector before assembling.

b.x.array[:] = 0.0
start = perf_counter()
dolfinx.fem.petsc.assemble_vector(b.x.petsc_vec, L_compiled)
b.x.scatter_reverse(dolfinx.la.InsertMode.add)
end = perf_counter()

We can now compare the performance of assembling over the linear mesh with the same resolution:

Assembly on linear grid#

Define the linear form

v_lin = ufl.TestFunction(V_lin)
f_lin = dolfinx.fem.Constant(linear_mesh, (0, -9.81))
dx_lin = ufl.Measure("dx", domain=linear_mesh)
L_lin = linear_form(f_lin, v_lin, dx_lin)
L_lin_compiled = dolfinx.fem.form(L_lin)

Define and assemble the vector

b_lin = dolfinx.fem.Function(V_lin)
b_lin.x.array[:] = 0.0
start_lin = perf_counter()
dolfinx.fem.petsc.assemble_vector(b_lin.x.petsc_vec, L_lin_compiled)
b_lin.x.scatter_reverse(dolfinx.la.InsertMode.add)
end_lin = perf_counter()

Comparison with curved assembly#

Linear time (b): 5.18e-03 Curved/Linear=3.53e+00

We additionally check the estimated quadrature degree for each integral

from ufl.algorithms import expand_derivatives, estimate_total_polynomial_degree

print(f"Curved (b) estimate: {estimate_total_polynomial_degree(expand_derivatives(L))}")
print(f"Linear (b) estimate: {estimate_total_polynomial_degree(expand_derivatives(L_lin))}")

Curved (b) estimate: 3
Linear (b) estimate: 3

We observe that the assembly time is faster on the linear mesh, even though the quadrature degree is the same for both integrals. This is due to the computation of the Jacobian inside the assembly kernel.

What about assembling a matrix?

Bilinear forms (matrices)#

As we have seen in The variational form of a projection, we additionally get a sparse matrix when we assemble a bilinear form?

Why is it a sparse matrix?

As the basis function \(\phi_i\) only have local support within the elements sharing the entity they are associated to, we only get a few non-zero entries in each row of the matrix.

We use a similar approach as above to assemble a matrix. We start by defining the bilinear form with a TestFunction and a TrialFunction.

We consider the following bi-linear form

\[ a(u,v) = \int_\Omega u \cdot v + \nabla u \cdot \nabla v ~\mathrm{d}x. \]

def bilinear_form(u, v, dx):
    return ufl.inner(u, v) * dx + ufl.inner(ufl.grad(u), ufl.grad(v)) * dx

a = bilinear_form(ufl.TrialFunction(V), v, dx_curved)
a_compiled = dolfinx.fem.form(a)

Next, as a sparse matrix is expensive to create, we use the dolfinx.fem.petsc.create_matrix() function to create a matrix which we can use multiple times if we want to assemble the matrix multiple times.

Accumulation of values when assembling a matrix

DOLFINx does not zero out existing values in the matrix when assembling, so if you assemble the matrix multiple times, the values will be added to the existing values. Call A.zeroEntries() to zero out the matrix before assembling.

A = dolfinx.fem.petsc.create_matrix(a_compiled)
A.zeroEntries()

Next we can assemble the matrix using dolfinx.fem.petsc.assemble_matrix()

start_A = perf_counter()
dolfinx.fem.petsc.assemble_matrix(A, a_compiled)
A.assemble()
end_A = perf_counter()

We do a similar computation for the linear mesh

a_lin = bilinear_form(ufl.TrialFunction(V_lin), v_lin, dx_lin)
a_lin_compiled = dolfinx.fem.form(a_lin)
A_lin = dolfinx.fem.petsc.create_matrix(a_lin_compiled)
A_lin.zeroEntries()
start_A_lin = perf_counter()
dolfinx.fem.petsc.assemble_matrix(A_lin, a_lin_compiled)
A_lin.assemble()
end_A_lin = perf_counter()

We compare the assembly times

Linear time (A): 3.78e-01 Curved/Linear=1.47e+00
Linear time (b): 5.18e-03 Curved/Linear=3.53e+00

We observe that assembling the matrix is two orders of magnitude slower than assembling the vector. We also observe that the assembly on a linear grid is faster than on a curved grid.

We also compare the estimated quadrature degree of the integrals

Curved (A) estimate: 6
Linear (A) estimate: 6