Minimal examples of the desired firedrake operations.

test_minimal.py assembles $\int v \text{ d}x$, expressed in UFL as v * dx. This is a 1-form and the first example we are interested in.

test_laplace.py assembles $\int \nabla v \cdot \nabla u \text{ d}x$, expressed in UFL as dot(grad(v), grad(u)) * dx. This is a 2-form and an example of the further forms we will be assembling. A detailed explanation of this form can be found in section 3.2 of this paper. It is assembled over two different finite element spaces - one for u and one for v. Note that the action of this 2-form against a fixed vector is used for checking - this is to avoid storing large matrices.

test_coefficients.py assembles $\int (f + g)v \text{ d}x$, expressed in UFL as (f + g) * v dx. f, g and v are all expressed in different finite element spaces. This demonstrates the use of coefficients.

Along with the full scripts, there are also *_no_firedrake.py versions, which can be run without Firedrake installed, so long as the corresponding data files are accessible.

stream_mfe.py notes the issues encountered using streams and einsums together.