# Assemble forms from mixed space formulation

**URL:** <https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505>\
**Category:** Errors\
**Created:** [June 10, 2022, 12:19pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505 "2022-06-10T12:19:14Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![MPenaR](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/mpenar/32/2985_2.png) [@MPenaR](https://fenicsproject.discourse.group/u/MPenaR)\
**Post date:** [June 10, 2022, 12:19pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505/1 "2022-06-10T12:19:15Z")

</div>

I have solved the Helmholtz equation with complex “wave number” in `dolfin` without problems by splitting it in its real and imaginary parts and using mixed function spaces.

The problem along with the code is already shown [here](https://fenicsproject.discourse.group/t/solution-does-not-fulfil-homogeneous-neumann-conditions/8493), but I copy here the code, just in case:

```python
#imports
from dolfin import RectangleMesh, MeshFunction, SubDomain, Measure, FunctionSpace, MixedFunctionSpace, Constant
from dolfin import TrialFunctions, TestFunctions, inner, grad, dx, Function, solve, plot, Point, near, Expression
from dolfin import assemble

#geometrical constants and mesh generation
Lx = 0.01
aspect_ratio = 100
Ly = aspect_ratio*Lx
Nx = 5
Ny = aspect_ratio*Nx 

mesh = RectangleMesh( Point(0,0), Point(Lx,Ly), Nx, Ny )

dim_edge = 1
def_value = 0

boundary_markers = MeshFunction("size_t", mesh, dim_edge, def_value)

tol = 1.0E-14

front = 1
back = 2

class Gamma_front(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], 0, tol)

GF = Gamma_front()
GF.mark(boundary_markers, front)

class Gamma_back(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], Lx, tol)

GB = Gamma_back()
GB.mark(boundary_markers, back)

ds = Measure('ds', domain=mesh, subdomain_data=boundary_markers)

#Physical constants
kappa = 200
h_conv = 15 
epsilon = 0.8
sigma = 5.67E-8
T_air = 298
rho = 2700
c= 900
omega = 0.8

alpha = 0.4

I = 600*(1 + 0.0j)

sx = -15*Lx
sy = 0.5*Ly

Q_R = Expression("alpha*I_R/(2*pi)*(x[0]-sx)/((x[0]-sx)*(x[0]-sx) + (x[1]-sy)*(x[1]-sy))", degree=2, alpha = alpha, I_R = I.real, sx=sx, sy=sy)
Q_I = Expression("alpha*I_I/(2*pi)*(x[0]-sx)/((x[0]-sx)*(x[0]-sx) + (x[1]-sy)*(x[1]-sy))", degree=2, alpha = alpha, I_I = I.imag, sx=sx, sy=sy)

# Funtion spaces 
P2 = FunctionSpace(mesh, "Lagrange", 1 )
H = MixedFunctionSpace( P2, P2 ) 
Phi_R, Phi_I = TrialFunctions(H)
Psi_R, Psi_I = TestFunctions(H)

# variational forms
a = kappa*( inner( grad(Phi_R), grad(Psi_R)) + inner( grad(Phi_I), grad(Psi_I) ))*dx \
                                          - rho*omega*c*( Phi_R*Psi_I - Phi_I*Psi_R)*dx \
                                          + h_eff*( Phi_R*Psi_R + Phi_I*Psi_I)*ds(front) \
                                          + h_eff*( Phi_R*Psi_R + Phi_I*Psi_I)*ds(back)
L = ( Q_R*Psi_R + Q_I*Psi_I )*ds(front)

#problem solution
T = Function(H)
solve(a==L, T)
T_R, T_I = T.split()

```

However, now I was interested in solving similar problems with different right hand sides, so I tried to compute the assembled matrix of the bilinear form using:

```auto
from dolfin import assemble
A = assemble(a)
b = assemble(L)

```

and I get the message error: `ArityMismatch: Adding expressions with non-matching form arguments ('v_0^0', 'v_1^1') vs ('v_0^1', 'v_1^0').`

After performing a look on the internet I saw that this can be caused by the forms not having all the trial and test functions of the mixed space, but I don’t understand how to fix it.

Thank you very much

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<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:** [June 10, 2022, 12:30pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505/2 "2022-06-10T12:30:06Z")

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> [@MPenaR](#):
>
> `MixedFunctionSpace`

Are you sure that you want to use `MixedFunctionSpace`, and not  
`H = FunctionSpace(mesh, MixedElement([P2, P2]))`?  
As far as im aware, `MixedFunctionSpace` is used for @cdaversin’s mixed dimensional FE implementation, and not to created a MixedElement on the same meshes

---

<div class="post-metadata">

**Author:** ![MPenaR](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/mpenar/32/2985_2.png) [@MPenaR](https://fenicsproject.discourse.group/u/MPenaR)\
**Post date:** [June 10, 2022, 12:33pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505/3 "2022-06-10T12:33:13Z")

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I’m sure I was not understanding what I was doing. I thought that both things were equivalent. I will rewrite it as you suggest. Is there any place where I can read about the difference between both?

Thanks!

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<div class="post-metadata">

**Author:** ![MPenaR](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/mpenar/32/2985_2.png) [@MPenaR](https://fenicsproject.discourse.group/u/MPenaR)\
**Post date:** [June 10, 2022, 12:39pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505/4 "2022-06-10T12:39:34Z")

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Rewritten, works now!

---

<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:** [June 10, 2022, 12:56pm UTC](https://fenicsproject.discourse.group/t/assemble-forms-from-mixed-space-formulation/8505/5 "2022-06-10T12:56:45Z")

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See for instance; [https://dl.acm.org/doi/abs/10.1145/3471138](https://dl.acm.org/doi/abs/10.1145/3471138)