Support -snes_adapt

[pbrubeck]

Description

Codex-assisted.

Implements UFL-level refinement of a variational problem and PETSc-level refinement of the SNES solver. In this way, users will not require to write the adaptivity loop themselves.

from firedrake import *

mesh = UnitSquareMesh(10, 10)
hierarchy = AdaptiveMeshHierarchy(mesh)

V = FunctionSpace(mesh, "CG", 1)
u = Function(V)
v = TestFunction(v_space)
F = inner(grad(u), grad(v)) * dx - f * v * dx

def marking_callback(solver_context, solution):
    # Select cells based on some criteria (e.g., Dorfler marking)
    marker = Function(DG0)
    ...
    return marker

params = {
    "snes_adapt_sequence": 5,  # Number of adaptation iterations
}

problem = NonlinearVariationalProblem(F, u)
solver = NonlinearVariationalSolver(problem, solver_parameters=params, 
                                    marking_callback=marking_callback)
solver.solve()          
u_adapt = solver.get_adapted_solution()

Depends on https://gitlab.com/petsc/petsc/-/merge_requests/9407

[pbrubeck]

Should we only keep alive the coarsest level?

[pbrubeck]

We should prolong instead, and that would be done at a later point

Suggested change

new.interpolate(expr)

[pbrubeck]

Suggested change

	manager.restrict(c, mapped)
	manager.restrict(c, mapped)
	else:
	mapped.interpolate(c)