# Mixed formulation of Maxwell's equations

**URL:** <https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814>\
**Category:** variational formulation\
**Created:** [March 9, 2022, 3:59pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814 "2022-03-09T15:59:12Z")\
**Posts on this page:** 20\
**Page:** 1

<div class="post-metadata">

**Author:** ![MatsievskiySV](https://avatars.discourse-cdn.com/v4/letter/m/439d5e/32.png) [@MatsievskiySV](https://fenicsproject.discourse.group/u/MatsievskiySV)\
**Post date:** [March 9, 2022, 3:59pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/1 "2022-03-09T15:59:12Z")

</div>

In [this paper](https://people.inf.ethz.ch/arbenz/icap2000.pdf) there’s a mixed formulation of Maxwells equations:

> Find (\lambda, e, p)\in\mathbb{R}\times H\_0(\operatorname{curl}; \Omega)\times H^1\_0 (\Omega) such that e \neq d and:  
> (a) (\operatorname{curl} e, \operatorname{curl} \psi) + (\operatorname{grad}p, \psi)=\lambda(e, \psi),\quad\forall\psi\in H\_0(\operatorname{curl}; \Omega)  
> (b) (e, \operatorname{grad} q)=0,\quad \forall q\in H\_0^1(\Omega)

I’m trying to implement it in DolfinX like so

```python
N1curl = ufl.FiniteElement("Nedelec 1st kind H(curl)", mesh.ufl_cell(), 1)
H1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V = dolfinx.FunctionSpace(mesh, ufl.MixedElement(N1curl, H1))

e, p = ufl.TrialFunctions(V)
psi, q = ufl.TestFunctions(V)

a = ufl.inner(ufl.curl(e), ufl.curl(psi)) * ufl.dx + ufl.inner(ufl.grad(p), psi) * ufl.dx
a += ufl.inner(e, ufl.grad(q)) * ufl.dx
b = ufl.inner(e, psi) * ufl.dx

```

but I get a ` Zero pivot in LU factorization` SLEPc error. What am I doing wrong?

Full code:

```python
import dolfinx
import dolfinx.io
import ufl
from mpi4py import MPI
from slepc4py import SLEPc
import numpy as np

mesh = dolfinx.RectangleMesh(
    MPI.COMM_WORLD,
    [np.array([0, 0, 0]), np.array([np.pi, np.pi/2, 0])], [40, 40],
    cell_type=dolfinx.cpp.mesh.CellType.triangle)

mesh.topology.create_connectivity(mesh.topology.dim-1, mesh.topology.dim)

N1curl = ufl.FiniteElement("Nedelec 1st kind H(curl)", mesh.ufl_cell(), 1)
H1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V = dolfinx.FunctionSpace(mesh, ufl.MixedElement(N1curl, H1))

e, p = ufl.TrialFunctions(V)
psi, q = ufl.TestFunctions(V)

a = ufl.inner(ufl.curl(e), ufl.curl(psi)) * ufl.dx + ufl.inner(ufl.grad(p), psi) * ufl.dx
a += ufl.inner(e, ufl.grad(q)) * ufl.dx
b = ufl.inner(e, psi) * ufl.dx

bc_facets = np.where(np.array(dolfinx.cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bc_dofs = dolfinx.fem.locate_dofs_topological(V, mesh.topology.dim-1, bc_facets)

u_bc = dolfinx.Function(V)
with u_bc.vector.localForm() as loc:
    loc.set(0)
bc = dolfinx.DirichletBC(u_bc, bc_dofs)

A = dolfinx.fem.assemble_matrix(a, bcs=[bc])
A.assemble()
B = dolfinx.fem.assemble_matrix(b, bcs=[bc])
B.assemble()

eps = SLEPc.EPS().create(MPI.COMM_WORLD)
eps.setOperators(A, B)
eps.setType(SLEPc.EPS.Type.KRYLOVSCHUR)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_MAGNITUDE)
eps.setPowerShiftType(SLEPc.EPS.PowerShiftType.CONSTANT)
eps.setTarget(2)
eps.setDimensions(4)
eps.solve()

vals = [(i, eps.getEigenvalue(i)) for i in range(eps.getConverged())]
vals.sort(key=lambda x: x[1].real)
print(vals)

j = 0
E = dolfinx.Function(V)
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "out.xdmf", "w") as xdmf:
    xdmf.write_mesh(mesh)
    for i, _ in vals:
        print(eps.getEigenpair(i, E.vector))
        E.name = f"E-{j}"
        xdmf.write_function(E)
        j += 1

```

---

<div class="post-metadata">

**Author:** ![nate](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/nate/32/17_2.png) [@nate](https://fenicsproject.discourse.group/u/nate)\
**Post date:** [March 10, 2022, 4:03am UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/2 "2022-03-10T04:03:08Z")

</div>

I know this doesn’t answer your question, but perhaps it’s a useful first step:

By the nature of the generalised eigenvalue problem you shouldn’t need to implement the involution term as you would for a time harmonic Maxwell problem where the frequency is not an eigenvalue of the system.

Consider the following. Let (u, \lambda \> 0) solve the generealised eigenvalue problem, then we have

\begin{align} \nabla \times \nabla \times u &= \lambda u, \\ \nabla \cdot (\nabla \times \nabla \times u) &= \lambda \nabla \cdot u, \\ 0 &= \nabla \cdot u. \end{align}

If you modify your problem do you have a full diagonal and as such does the eigensolver converge?

Also there’s a very old (but excellent) demo [here](https://fenicsproject.org/olddocs/dolfin/latest/python/demos/maxwell-eigenvalues/demo_maxwell-eigenvalues.py.html) which may help.

---

<div class="post-metadata">

**Author:** ![MatsievskiySV](https://avatars.discourse-cdn.com/v4/letter/m/439d5e/32.png) [@MatsievskiySV](https://fenicsproject.discourse.group/u/MatsievskiySV)\
**Post date:** [March 10, 2022, 3:15pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/3 "2022-03-10T15:15:20Z")

</div>

Yes, I’ve seen this example. For some reason I don’t get the same eigenvalues as in the example.

According to [this paper](https://doi.org/10.1016/0045-7825(87)90053-3), presented in the question formulation, should better impose divergence free condition \nabla\cdot u = 0.

```nohighlight
import dolfinx
import dolfinx.io
import ufl
from mpi4py import MPI
from slepc4py import SLEPc
import numpy as np

mesh = dolfinx.RectangleMesh(
    MPI.COMM_WORLD,
    [np.array([0, 0, 0]), np.array([np.pi, np.pi, 0])], [40, 40],
    cell_type=dolfinx.cpp.mesh.CellType.triangle)

mesh.topology.create_connectivity(mesh.topology.dim-1, mesh.topology.dim)

N1curl = ufl.FiniteElement("Nedelec 1st kind H(curl)", mesh.ufl_cell(), 1)
V = dolfinx.FunctionSpace(mesh, N1curl)

u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)

a = ufl.inner(ufl.curl(u), ufl.curl(v)) * ufl.dx + ufl.inner(ufl.curl(u), ufl.curl(v)) * ufl.dx
b = ufl.inner(u, v) * ufl.dx

bc_facets = np.where(np.array(dolfinx.cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bc_dofs = dolfinx.fem.locate_dofs_topological(V, mesh.topology.dim-1, bc_facets)

u_bc = dolfinx.Function(V)
with u_bc.vector.localForm() as loc:
    loc.set(0)
bc = dolfinx.DirichletBC(u_bc, bc_dofs)

A = dolfinx.fem.assemble_matrix(a, bcs=[bc])
A.assemble()
B = dolfinx.fem.assemble_matrix(b, bcs=[bc])
B.assemble()

eps = SLEPc.EPS().create(MPI.COMM_WORLD)
eps.setOperators(A, B)
eps.setType(SLEPc.EPS.Type.KRYLOVSCHUR)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_MAGNITUDE)
eps.setPowerShiftType(SLEPc.EPS.PowerShiftType.CONSTANT)
eps.setTarget(5.5)
eps.setDimensions(12)
eps.solve()

np.isclose(eps.getEigenvalue(i), 1)]
vals = [(i, eps.getEigenvalue(i)) for i in range(eps.getConverged())]

vals.sort(key=lambda x: x[1].real)

j = 0
E = dolfinx.Function(V)
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "out.xdmf", "w") as xdmf:
    xdmf.write_mesh(mesh)
    for i, _ in vals:
        print(eps.getEigenpair(i, E.vector))
        E.name = f"E-{j}"
        xdmf.write_function(E)
        j += 1

```

---

<div class="post-metadata">

**Author:** ![MatsievskiySV](https://avatars.discourse-cdn.com/v4/letter/m/439d5e/32.png) [@MatsievskiySV](https://fenicsproject.discourse.group/u/MatsievskiySV)\
**Post date:** [March 11, 2022, 6:25pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/4 "2022-03-11T18:25:49Z")

</div>

> For some reason I don’t get the same eigenvalues as in the example.

Wops, there’s a typo `a = ufl.inner(ufl.curl(u), ufl.curl(v)) * ufl.dx`. Example works fine.

---

<div class="post-metadata">

**Author:** ![MatsievskiySV](https://avatars.discourse-cdn.com/v4/letter/m/439d5e/32.png) [@MatsievskiySV](https://fenicsproject.discourse.group/u/MatsievskiySV)\
**Post date:** [March 17, 2022, 2:22pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/5 "2022-03-17T14:22:10Z")

</div>

Turns out, that the problem was not with the formulation, but with the configuration of the Spectral Transformation (ST) if SLEPc. It shold be configured as follows:

```python
st = eps.getST()
st.setType(SLEPc.ST.Type.SINVERT)

```

Checking \int\_{\Omega}\psi\;\text{d}x values I can see that unphysical modes with \lambda=1 all have high integral value:

```auto
eigenvalue: 0.999689889049; int(psi): -6.308888402483e-14
eigenvalue: 0.999967476453; int(psi): -1.645503611179e-12
eigenvalue: 1.000000000000; int(psi): 1.337930720732e-03
eigenvalue: 1.000000000000; int(psi): -1.598505351224e-03
eigenvalue: 1.000000000000; int(psi): 2.309336061885e-02
eigenvalue: 1.000000000000; int(psi): -1.404544335133e-03
eigenvalue: 1.000000000000; int(psi): 3.885881138543e-04
eigenvalue: 1.000000000000; int(psi): 6.228103117720e-04
eigenvalue: 1.000000000000; int(psi): 1.856027911829e-03
eigenvalue: 1.000000000000; int(psi): 3.188035050973e-03
eigenvalue: 2.000342166381; int(psi): 4.142071959184e-14
eigenvalue: 3.997258892126; int(psi): -4.388551693095e-14
eigenvalue: 3.997260387796; int(psi): -1.283612474464e-14
eigenvalue: 4.997207026787; int(psi): 7.494442252868e-15
eigenvalue: 5.002446610361; int(psi): 1.745873985960e-14
eigenvalue: 8.005430745727; int(psi): -2.022234716329e-14
eigenvalue: 8.984888327100; int(psi): 4.700200964440e-14
eigenvalue: 8.987372947191; int(psi): -5.860825488704e-14
eigenvalue: 9.992103624262; int(psi): -2.649171526890e-14
eigenvalue: 9.992163510773; int(psi): 1.579640268136e-14

```

Full DolfinX code:

```python
import dolfinx
import dolfinx.io
import ufl
from mpi4py import MPI
from slepc4py import SLEPc
import numpy as np

mesh = dolfinx.RectangleMesh(
    MPI.COMM_WORLD,
    [np.array([0, 0, 0]), np.array([np.pi, np.pi, 0])], [40, 40],
    cell_type=dolfinx.cpp.mesh.CellType.triangle)

mesh.topology.create_connectivity(mesh.topology.dim-1, mesh.topology.dim)

N1curl = ufl.FiniteElement("Nedelec 1st kind H(curl)", mesh.ufl_cell(), 1)
H1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V = dolfinx.FunctionSpace(mesh, ufl.MixedElement(N1curl, H1))

e, p = ufl.TrialFunctions(V)
psi, q = ufl.TestFunctions(V)

a = ufl.inner(ufl.curl(e), ufl.curl(psi)) * ufl.dx + ufl.inner(ufl.grad(p), psi) * ufl.dx
a += ufl.inner(e, ufl.grad(q)) * ufl.dx
b = ufl.inner(e, psi) * ufl.dx

bc_facets = np.where(np.array(dolfinx.cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bc_dofs = dolfinx.fem.locate_dofs_topological(V, mesh.topology.dim-1, bc_facets)

u_bc = dolfinx.Function(V)
with u_bc.vector.localForm() as loc:
    loc.set(0)
bc = dolfinx.DirichletBC(u_bc, bc_dofs)

A = dolfinx.fem.assemble_matrix(a, bcs=[bc])
A.assemble()
B = dolfinx.fem.assemble_matrix(b, bcs=[bc])
B.assemble()

eps = SLEPc.EPS().create(MPI.COMM_WORLD)
eps.setOperators(A, B)
eps.setType(SLEPc.EPS.Type.KRYLOVSCHUR)
st = eps.getST()
st.setType(SLEPc.ST.Type.SINVERT)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_MAGNITUDE)
eps.setTarget(5.5)
eps.setDimensions(20)
eps.solve()

vals = [(i, eps.getEigenvalue(i)) for i in range(eps.getConverged())]
vals.sort(key=lambda x: x[1].real)

j = 0
E = dolfinx.Function(V)
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "out.xdmf", "w") as xdmf:
    xdmf.write_mesh(mesh)
    for i, _ in vals:
        E1, E2 = E.split()
        print(f"eigenvalue: {eps.getEigenpair(i, E.vector).real:.12f};\t"
              f"int(psi): {dolfinx.fem.assemble_scalar(E2 * ufl.dx):.12e}")
        E1.name = f"E1-{j:03d}-{eps.getEigenpair(i, E.vector).real:.4f}"
        E2.name = f"E2-{j:03d}-{eps.getEigenpair(i, E.vector).real:.4f}"
        xdmf.write_function(E1)
        xdmf.write_function(E2)
        j += 1

```

---

<div class="post-metadata">

**Author:** ![philippweder](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/philippweder/32/4462_2.png) [@philippweder](https://fenicsproject.discourse.group/u/philippweder)\
**Post date:** [December 12, 2022, 1:07pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/6 "2022-12-12T13:07:05Z")

</div>

I am doing something similar and I was wondering what the explanation for the unphysical modes with \lambda = 1 was?

---

<div class="post-metadata">

**Author:** ![philippweder](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/philippweder/32/4462_2.png) [@philippweder](https://fenicsproject.discourse.group/u/philippweder)\
**Post date:** [December 14, 2022, 4:14pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/7 "2022-12-14T16:14:45Z")

</div>

If you zero out the rows corresponding to the boundary conditions in the operator `B`, you actually do not retrieve spurious eigenmodes with eigenvalue 1.0, i.e.

```auto
B = assemble_matrix(b, bcs=[bc], diagonal=0.0)

```

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 1:31pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/8 "2023-05-03T13:31:51Z")

</div>

Hi, when I want to run this program, I get the error "AttributeError: module ‘dolfinx’ has no attribute ‘RectangleMesh’ "… Do you have any idea why? Thanks…

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [May 3, 2023, 1:44pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/9 "2023-05-03T13:44:39Z")

</div>

The code here is written for an older version of DOLFINx.  
`dolfinx.RectangleMesh` has been replaced with `dolfinx.mesh.create_rectangle`.

Similarly, You should replace

> [@MatsievskiySV](#):
>
> `bc = dolfinx.DirichletBC(u_bc, bc_dofs)`

with

```python
dolfinx.fem.dirichletbc

```

and

> [@MatsievskiySV](#):
>
> ```auto
> A = dolfinx.fem.assemble_matrix(a, bcs=[bc])
> A.assemble()
> B = dolfinx.fem.assemble_matrix(b, bcs=[bc])
> B.assemble()
> 
> ```

with

```auto
A = dolfinx.fem.petsc.assemble_matrix(a, bcs=[bc])
A.assemble()
B = dolfinx.fem.petsc.assemble_matrix(b, bcs=[bc])
B.assemble()

```

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 4:28pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/10 "2023-05-03T16:28:06Z")

</div>

Thank you very much for your help. In the meantime, I tried to reinstall everything related Dolfinx. Then I followed your fixes. Now I get the error " ModuleNotFoundError: No module named ‘basix.ufl\_wrapper’ ". Searched for it but couldn’t find anything. Could you possibly help please?

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [May 3, 2023, 4:41pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/11 "2023-05-03T16:41:36Z")

</div>

This probably has to do with having non-compatible versions of Basix, ffcx and dolfinx. How did you install each individual component?

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 4:55pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/12 "2023-05-03T16:55:45Z")

</div>

I tried many things. spack installation, ubuntu ppa, and individually (e.g., pip for ffcx). My system is Ubuntu 22.04.2 LTS. Can you suggest the best way to install everything as simple as it can be?

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [May 3, 2023, 5:05pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/13 "2023-05-03T17:05:12Z")

</div>

The problem is if you have mixed all of these

> [@metin\_demir](#):
>
> I tried many things. spack installation, ubuntu ppa, and individually (e.g., pip for ffcx). My system is Ubuntu 22.04.2 LTS

There can be many conflicting versions/dependencies from the different installation methods.

The easiest is to use `conda`.  
Second easiest is to use `docker`  
Third easiest is spack  
Fourth easiest is ubuntu PPA(But not recommended as the package there is not up to date)

However, i cannot help you any further if you dont post a minimal code example that yields an error, and the full trace of the error message.

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 5:08pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/14 "2023-05-03T17:08:28Z")

</div>

Many thanks. I will try reinstallation using conda. If the next try fails too, I will post the code with the error message.

---

<div class="post-metadata">

**Author:** ![MatsievskiySV](https://avatars.discourse-cdn.com/v4/letter/m/439d5e/32.png) [@MatsievskiySV](https://fenicsproject.discourse.group/u/MatsievskiySV)\
**Post date:** [May 3, 2023, 5:27pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/15 "2023-05-03T17:27:17Z")

</div>

If you decide to build everything manually (e.g. for the computational cluster), here’s a semi-automatic script

```bash
SOURCE_DIR=$(realpath .)
BUILD_DIR=$SOURCE_DIR/build-dir
INSTALL_DIR=/opt/dolfinx

pip3 uninstall -y --break-system-packages fenics-dolfinx fenics-ffcx fenics-ufl fenics-basix

sudo apt install libpetsc-complex-dev libslepc-complex-dev python3-petsc4py-complex python3-slepc4py-complex

BASIX_COMMIT=185dd46f4b0121539fbc126ed338836f453b8c17
DOLFIN_COMMIT=3784623b46081f408052b42123bb290da158a3e9
UFL_COMMIT=6eeef938c877634dcb7c8acb101bb2363ed1455b
FFCX_COMMIT=c1dc25e7e6ec2661743449a27906f203760c0627

git clone --depth 1 https://github.com/KarypisLab/GKlib.git
git clone --depth 1 https://github.com/KarypisLab/METIS.git
git clone --depth 1 https://github.com/KarypisLab/ParMETIS.git $SOURCE_DIR/ParMETIS
git clone --depth 1 https://github.com/ornladios/ADIOS2.git $SOURCE_DIR/ADIOS2
git clone https://github.com/FEniCS/basix.git $SOURCE_DIR/basix
git -C $SOURCE_DIR/basix checkout $BASIX_COMMIT
git clone https://github.com/FEniCS/dolfinx.git $SOURCE_DIR/dolfinx
git -C $SOURCE_DIR/dolfinx checkout $DOLFINX_COMMIT
git clone https://github.com/FEniCS/ufl.git $SOURCE_DIR/ufl
git -C $SOURCE_DIR/ufl checkout $UFL_COMMIT
git clone https://github.com/FEniCS/ffcx.git $SOURCE_DIR/ffcx
git -C $SOURCE_DIR/ffcx checkout $FFCX_COMMIT

scalar_type=complex

CMAKE_PREFIX_PATH="$CMAKE_PREFIX_PATH:$INSTALL_PATH/lib/cmake/:$INSTALL_DIR/share/cmake/"
PETSC_DIR="/usr/lib/petscdir/petsc-$scalar_type"
SLEPC_DIR="/usr/lib/slepcdir/slepc-$scalar_type"
PETSC_ARCH="linux-gnu-$scalar_type-64"
PYTHONPATH="$PETSC_DIR/lib/python3/dist-packages:$SLEPC_DIR/lib/python3/dist-packages"
PKG_CONFIG_PATH="$PETSC_DIR/lib/pkgconfig:$PKG_CONFIG_PATH"
LD_LIBRARY_PATH="$PETSC_DIR/lib:$LD_LIBRARY_PATH"

# GKlib
pushd GKlib
make config prefix=$INSTALL_DIR
sudo make install
popd

# METIS
pushd METIS
make config prefix=$INSTALL_DIR
sudo make install
popd

# ParMETIS
pushd ParMETIS
make config cc=mpicc prefix=$INSTALL_DIR
sudo make install
popd

# ADIOS2
rm -rf $BUILD_DIR
mkdir -p $BUILD_DIR

cmake -DCMAKE_BUILD_TYPE=Release -DCMAKE_INSTALL_PREFIX=$INSTALL_DIR \
      -DADIOS2_BUILD_EXAMPLES=OFF -DBUILD_TESTING=OFF \
      -B $BUILD_DIR -S $SOURCE_DIR/ADIOS2
cmake --build $BUILD_DIR -j$(nproc)
sudo cmake --install $BUILD_DIR

# Basix
rm -rf $BUILD_DIR
mkdir -p $BUILD_DIR

cmake -DCMAKE_BUILD_TYPE=Release -DCMAKE_INSTALL_PREFIX=$INSTALL_DIR \
      -B $BUILD_DIR -S $SOURCE_DIR/basix/cpp
cmake --build $BUILD_DIR -j$(nproc)
sudo cmake --install $BUILD_DIR

pushd $SOURCE_DIR/basix
pip3 install --break-system-packages --user .
popd

# UFL
pushd $SOURCE_DIR/ufl
pip3 install --break-system-packages --user .
popd

# FFCX
pushd $SOURCE_DIR/ffcx
pip3 install --break-system-packages --user .
popd

# DolfinX
rm -rf $BUILD_DIR
mkdir -p $BUILD_DIR

cmake -DCMAKE_BUILD_TYPE=Release -DCMAKE_INSTALL_PREFIX=$INSTALL_DIR \
       -B $BUILD_DIR -S $SOURCE_DIR/dolfinx/cpp
cmake --build $BUILD_DIR -j$(nproc)
sudo cmake --install $BUILD_DIR
source $INSTALL_DIR/lib/dolfinx/dolfinx.conf

pushd $SOURCE_DIR/dolfinx/python
pip3 install --break-system-packages --user .
popd

```

Note, that the same project tags are not necessarily compatible, that’s why I use commit hashes.

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 8:07pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/16 "2023-05-03T20:07:55Z")

</div>

Thank you very much. I will try this as well.

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 8:54pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/17 "2023-05-03T20:54:31Z")

</div>

Sorry again for disturbing. The above code with your fixes (given below) now results in the error:

```auto
in <module>
    mesh = dolfinx.mesh.create_rectangle(
  File "/usr/lib/petsc/lib/python3/dist-packages/dolfinx/mesh.py", line 372, in create_rectangle
    mesh = _cpp.mesh.create_rectangle(comm, points, n, cell_type, ghost_mode, partitioner, diagonal)
TypeError: create_rectangle(): incompatible function arguments. The following argument types are supported:
    1. (comm: MPICommWrapper, p: List[List[float[2]][2]], n: List[int[2]], celltype: dolfinx.cpp.mesh.CellType, ghost_mode: dolfinx.cpp.mesh.GhostMode, partitioner: Callable[[MPICommWrapper, int, int, dolfinx::graph::AdjacencyList<long>, dolfinx.cpp.mesh.GhostMode], dolfinx::graph::AdjacencyList<int>], diagonal: dolfinx.cpp.mesh.DiagonalType) -> dolfinx.cpp.mesh.Mesh

Invoked with: <mpi4py.MPI.Intracomm object at 0x7ff8947af1e0>, [array([0, 0, 0]), array([3.14159265, 3.14159265, 0.])], [40, 40], <CellType.triangle: 3>, <GhostMode.shared_facet: 1>, <built-in method of PyCapsule object at 0x7ff89232c1e0>, <DiagonalType.right: 1>

```

The code is exactly this:

```auto
import dolfinx
import dolfinx.io
import ufl
from mpi4py import MPI
from slepc4py import SLEPc
import numpy as np

mesh = dolfinx.mesh.create_rectangle(
    MPI.COMM_WORLD,
    [np.array([0, 0, 0]), np.array([np.pi, np.pi, 0])], [40, 40],
    cell_type=dolfinx.cpp.mesh.CellType.triangle)

mesh.topology.create_connectivity(mesh.topology.dim-1, mesh.topology.dim)

N1curl = ufl.FiniteElement("Nedelec 1st kind H(curl)", mesh.ufl_cell(), 1)
H1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V = dolfinx.FunctionSpace(mesh, ufl.MixedElement(N1curl, H1))

e, p = ufl.TrialFunctions(V)
psi, q = ufl.TestFunctions(V)

a = ufl.inner(ufl.curl(e), ufl.curl(psi)) * ufl.dx + ufl.inner(ufl.grad(p), psi) * ufl.dx
a += ufl.inner(e, ufl.grad(q)) * ufl.dx
b = ufl.inner(e, psi) * ufl.dx

bc_facets = np.where(np.array(dolfinx.cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bc_dofs = dolfinx.fem.locate_dofs_topological(V, mesh.topology.dim-1, bc_facets)

u_bc = dolfinx.Function(V)
with u_bc.vector.localForm() as loc:
    loc.set(0)
bc = dolfinx.fem.dirichletbc(u_bc, bc_dofs)

A = dolfinx.fem.petsc.assemble_matrix(a, bcs=[bc])
A.assemble()
B = dolfinx.fem.petsc.assemble_matrix(b, bcs=[bc])
B.assemble()

eps = SLEPc.EPS().create(MPI.COMM_WORLD)
eps.setOperators(A, B)
eps.setType(SLEPc.EPS.Type.KRYLOVSCHUR)
st = eps.getST()
st.setType(SLEPc.ST.Type.SINVERT)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_MAGNITUDE)
eps.setTarget(5.5)
eps.setDimensions(20)
eps.solve()

vals = [(i, eps.getEigenvalue(i)) for i in range(eps.getConverged())]
vals.sort(key=lambda x: x[1].real)

j = 0
E = dolfinx.Function(V)
with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "out.xdmf", "w") as xdmf:
    xdmf.write_mesh(mesh)
    for i, _ in vals:
        E1, E2 = E.split()
        print(f"eigenvalue: {eps.getEigenpair(i, E.vector).real:.12f};\t"
              f"int(psi): {dolfinx.fem.assemble_scalar(E2 * ufl.dx):.12e}")
        E1.name = f"E1-{j:03d}-{eps.getEigenpair(i, E.vector).real:.4f}"
        E2.name = f"E2-{j:03d}-{eps.getEigenpair(i, E.vector).real:.4f}"
        xdmf.write_function(E1)
        xdmf.write_function(E2)
        j += 1

```

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [May 3, 2023, 8:56pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/18 "2023-05-03T20:56:52Z")

</div>

> [@metin\_demir](#):
>
> `array([0, 0, 0]), array([3.14159265, 3.14159265, 0.])`

Change this to  
`array([0, 0]), array([3.14159265, 3.14159265])`

---

<div class="post-metadata">

**Author:** ![metin\_demir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/metin_demir/32/5005_2.png) [@metin\_demir](https://fenicsproject.discourse.group/u/metin_demir)\
**Post date:** [May 3, 2023, 9:02pm UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/19 "2023-05-03T21:02:24Z")

</div>

The error now is:  
V = dolfinx.FunctionSpace(mesh, ufl.MixedElement(N1curl, H1))  
AttributeError: module ‘dolfinx’ has no attribute ‘FunctionSpace’

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [May 4, 2023, 4:49am UTC](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814/20 "2023-05-04T04:49:07Z")

</div>

`dolfinx.fem.FunctionSpace`.  
I would strongly suggest that you familiarize yourself with the new api either by looking at the demos  
[https://docs.fenicsproject.org/dolfinx/v0.6.0/python/demos.html](https://docs.fenicsproject.org/dolfinx/v0.6.0/python/demos.html)  
or the tutorial  
[https://jsdokken.com/dolfinx-tutorial/](https://jsdokken.com/dolfinx-tutorial/)

[Next page](https://fenicsproject.discourse.group/t/mixed-formulation-of-maxwells-equations/7814.md?page=2)