Using Function Spaces with Varying Polynomial Degrees for hp-FEM in FEniCS(X)

Dr.Gardiner · November 23, 2025, 12:46am

Is it possible to have function spaces with spatially varying polynomial degrees, similar to what is used in hp-refinement (hp-FEM)? Are they implemented in FeNiCS(X)? If yes, are there any set of examples for getting started? If no, what are the alternative viable solutions?

dokken · November 24, 2025, 12:41pm

A strong enforcement of the continuity condition between different P-refinements are currently not available in FEniCSx. However, you can use a weak enforcement (DG/nitsche type).
I have for instance done this at:

github.com/jorgensd/presentations

fenicsx_pdesoft/logo_dg.py

main


# # Create a DG scheme coupling two subdomains with different function spaces
# # using interior penalty method (Nitsche's method) on the interface.
# Authors: Jørgen S. Dokken and Joseph P. Dean
# SPDX-License-Identifier: MIT

from mpi4py import MPI
import dolfinx
import dolfinx.fem.petsc
import ufl
import numpy as np
import numpy.typing as npt
import scifem


with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "logo.xdmf", "r") as xdmf:
    mesh = xdmf.read_mesh(ghost_mode=dolfinx.mesh.GhostMode.shared_facet)
    ct = xdmf.read_meshtags(mesh, name="cell_tags")
    mesh.topology.create_connectivity(mesh.topology.dim-1, mesh.topology.dim)
    ft = xdmf.read_meshtags(mesh, name="facet_tags")

This file has been truncated. show original

where I couple a P4 space and a P2 space with a DG method across the interfaces, resulting in the following figure:

Topic		Replies	Views
Spaces with varying polynomial degree (hp-refinement)	2	491	April 19, 2022
Polynomial level of function space changes the grid? General boundary-conditions , dolfinx	0	23	October 4, 2024
Could different Functionspaces be defined on different subdomains in one mesh? General legacy-dolfin	9	218	February 5, 2025
Different subdomains and different function spaces!	2	678	April 8, 2019
Enforcing weak continuity at interface for a mixed dimensional problem General	14	1338	December 4, 2024

Using Function Spaces with Varying Polynomial Degrees for hp-FEM in FEniCS(X)

Related topics