L2 projection of fine mesh solution to a coarse mesh

dolfinx
6

At some point you have to evaulate the function at the quadrature points of the coarse grid when doing the projection. The following code illlustrates how to compute the L2 projection from a fine grid to a coarse grid by:

  1. Get values of the fine function at the quadrature points of the coarse grid.
  2. Then do the projection
# Illustration of projection from one mesh to another
# Author: Jørgen S. Dokken
# SPDX License: MIT
import ufl
import dolfinx
from mpi4py import MPI
import basix.ufl
import dolfinx.fem.petsc
mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 10, 10)

# Compute function on fine mesh
V = dolfinx.fem.functionspace(mesh, (("Lagrange", 1)))
uh = dolfinx.fem.Function(V)
uh.interpolate(lambda x: x[0]+2*x[1])

# Create coarse mesh
coarser_mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 5, 5)
V_coarse = dolfinx.fem.functionspace(coarser_mesh, ("Lagrange", 1))


# Compute function for fine grid at quadrature points of the coarse grid
degree = 4
Qe = basix.ufl.quadrature_element(
    coarser_mesh.topology.cell_name(), degree=degree)
V_quadrature = dolfinx.fem.functionspace(coarser_mesh, Qe)
q_e = dolfinx.fem.Function(V_quadrature)
nmmid = dolfinx.fem.create_nonmatching_meshes_interpolation_data(V_quadrature.mesh._cpp_object,
                                                                 V_quadrature.element,
                                                                 V.mesh._cpp_object, padding=1e-5)

q_func = dolfinx.fem.Function(V_quadrature)
q_func.interpolate(uh, nmm_interpolation_data=nmmid)

# Project fine function at quadrature points to coarse grid
u = ufl.TrialFunction(V_coarse)
v = ufl.TestFunction(V_coarse)
a_coarse = ufl.inner(u, v) * ufl.dx
L_coarse = ufl.inner(q_func, v)*ufl.dx
problem = dolfinx.fem.petsc.LinearProblem(a_coarse, L_coarse)
u_coarse = problem.solve()

with dolfinx.io.VTXWriter(coarser_mesh.comm, "u_coarse.bp", [u_coarse], engine="BP4") as bp:
    bp.write(0.0)
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