Make tests basis-independent

[pbrubeck]

---

name: "Make tests basis-independent"
description: ''
title: ''
labels: ''
assignees: ''

---

Description

We need to drop the first commit and re-run CI with the equispaced variant.

A few observations:

1. VertexLimiter requires functions in Broken(mesh.coordinates.function_space())
2. Some tests like the one for EquationBC use a periodic sine wave manufactured solution, this could show the asymptotic convergence rate sooner with equispaced points (these points have the best approximation properties for periodic functions).
3. The tests for the adjoint of cross-mesh interpolation were testing for inequality of the output against the exact target Cofunction, and the source Function. The issue with this is that the absolute tolerance used in the dual comparison was the same used for primal comparsion, and the change to the spectral basis was changing the magnitude of the coefficients of the Cofunctions below the absolute tolerance. These inequality tests are quite meaningless and should be replaced with the same approach used in the multigrid restrict tests. We now test that <Ix, y> == <x, adjoint(I)y> for a choice of x \in V_src and y \in V_dest*.

Associated Pull Requests:

• Make a list (with links!)
• Like this PR#691

Fixes Issues:

• Make a list of issues (with links!)
• Like this Issue#2824

If issues are fixed by this PR, prepend each of them with the word
"fixes", so they are automatically closed when this PR is merged. For
example "fixes #123, fixes #456".

Checklist for author:

• I have linted the codebase (make lint in the firedrake source directory).
• My changes generate no new warnings.
• All of my functions and classes have appropriate docstrings.
• I have commented my code where its purpose may be unclear.
• I have included and updated any relevant documentation.
• Documentation builds locally (make linkcheck; make html; make latexpdf in firedrake/docs directory)
• I have added tests specific to the issues fixed in this PR.
• I have added tests that exercise the new functionality I have introduced
• Tests pass locally (pytest tests in the firedrake source directory) (useful, but not essential if you don't have suitable hardware).
• I have performed a self-review of my own code using the below guidelines.

Checklist for reviewer:

• Docstrings present
• New tests present
• Code correctly commented
• No bad "code smells"
• No issues in parallel
• No CI issues (excessive parallelism/memory usage/time/warnings generated)
• Upstream/dependent branches and PRs are ready

Feel free to add reviewers if you know there is someone who is already aware of this work.

Please open this PR initially as a draft and mark as ready for review once CI tests are passing.

[dham]

Just one trivial change.

[ReubenHill]

This looks good to go now