# How to define a time-dependent Neumann boundary condition if I have known the exact solution

**URL:** <https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906>\
**Category:** variational formulation\
**Tags:** dolfinx\
**Created:** [June 6, 2024, 10:40am UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906 "2024-06-06T10:40:33Z")\
**Posts on this page:** 6\
**Page:** 1

<div class="post-metadata">

**Author:** ![OsbertWang](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/osbertwang/32/2138_2.png) [@OsbertWang](https://fenicsproject.discourse.group/u/OsbertWang)\
**Post date:** [June 6, 2024, 10:40am UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/1 "2024-06-06T10:40:33Z")

</div>

Hello,

I want to solve a heat equation like this:

\frac{\partial u}{\partial t} - \nabla \cdot (\kappa \nabla u) = f

with the Neumann boundary condition:

-\frac{\partial u}{\partial \boldsymbol{n}} = g

I want to compute the error between exact solution and FEM solution.  
I noticed [https://jsdokken.com/dolfinx-tutorial/chapter3/robin\_neumann\_dirichlet.html](https://jsdokken.com/dolfinx-tutorial/chapter3/robin_neumann_dirichlet.html) which reads

```auto
u_ex = lambda x: 1 + x[0]**2 + 2*x[1]**2
x = SpatialCoordinate(mesh)
n = FacetNormal(mesh)
g = -dot(n, grad(u_ex(x)))

```

Here is my question:  
My exact solution is time-dependent, e.g. u\_{ex} = te^{-x-y}. How to define g as the tutorial.

My code is

```auto
from mpi4py import MPI
import numpy as np
from petsc4py import PETSc
from dolfinx.mesh import create_unit_square
from dolfinx.fem import functionspace
from ufl import dot, FacetNormal, grad, SpatialCoordinate

t = 0.0

domain = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = functionspace(domain, ("Lagrange", 1))

class u_ExactSolution():
    def __init__ (self, t):
        self.t = t

    def __call__ (self, x):
        value = np.zeros((2, x.shape[1]), PETSc.ScalrType)
        value[0] = self.t * np.exp(-x[0] - x[1])
        return value

u_ex = u_ExactSolution(t)

g = -dot(FacetNormal(domain), grad(u_ex(SpatialCoordinate(domain))))

```

Of course, it fails since ‘SpatialCoordinate’ object has no attribute ‘shape’

---

<div class="post-metadata">

**Author:** ![francesco-ballarin](https://avatars.discourse-cdn.com/v4/letter/f/f05b48/32.png) [@francesco-ballarin](https://fenicsproject.discourse.group/u/francesco-ballarin)\
**Post date:** [June 6, 2024, 10:59am UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/2 "2024-06-06T10:59:55Z")

</div>

If I were you, I’d rather do the math on paper (or with some symbolic algebra system) and compute myself -\partial u\_{ex} / \partial \mathbf{n}. It will be a math expression depending on x and y, and once you have it, you can type it up using `ufl.SpatialCoordinate` for (x, y).

This may be known as “method of manufactured solutions” in some textbooks.

---

<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 6, 2024, 11:19am UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/3 "2024-06-06T11:19:32Z")

</div>

You can automate the solution with something along the lines of

```python
t_c = dolfinx.fem.Constant(domain, 0.0)
x = ufl.SpatialCoordinate(mesh)
u_ex = t_c * ufl.exp(-x-y)

```

then if you want to update `t_c`, you can do that with  
`t_c.value += float(dt)`

---

<div class="post-metadata">

**Author:** ![OsbertWang](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/osbertwang/32/2138_2.png) [@OsbertWang](https://fenicsproject.discourse.group/u/OsbertWang)\
**Post date:** [June 6, 2024, 11:50am UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/4 "2024-06-06T11:50:42Z")

</div>

Thanks, @francesco-ballarin. I have already computed these terms on paper, and I think I should define different markers for each boundary.

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<div class="post-metadata">

**Author:** ![OsbertWang](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/osbertwang/32/2138_2.png) [@OsbertWang](https://fenicsproject.discourse.group/u/OsbertWang)\
**Post date:** [June 6, 2024, 12:04pm UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/5 "2024-06-06T12:04:28Z")

</div>

Thank you very much!  
Here is the MWE:

```auto
from mpi4py import MPI
from dolfinx.mesh import create_unit_square
from dolfinx.fem import functionspace
domain = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = functionspace(domain, ("Lagrange", 1))

from dolfinx.fem import Constant
from ufl import SpatialCoordinate, exp
t = 0.0
t_c = Constant(domain, t)
x = SpatialCoordinate(domain)
u_exact = t_c * exp(-x[0] - x[1])

from ufl import dot, FacetNormal, grad
n = FacetNormal(domain)
g = -dot(n, grad(u_exact))

```

Moreover, I tried another method whose MWE is

```auto
from mpi4py import MPI
import numpy as np
from petsc4py import PETSc
from dolfinx.mesh import create_unit_square
from dolfinx.fem import functionspace, Function
from ufl import dot, FacetNormal, grad

t = 0.0

domain = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = functionspace(domain, ("Lagrange", 1))

class u_ExactSolution():
    def __init__ (self, t):
        self.t = t

    def __call__ (self, x):
        return self.t * np.exp(-x[0] - x[1])

u_exact = Function(V)
u_ex = u_ExactSolution(t)
u_exact.interpolate(u_ex)

```

I remember that the tutorial says two methods, one is `SpatialCoordinate` and one is `interpolate`.

---

<div class="post-metadata">

**Author:** ![francesco-ballarin](https://avatars.discourse-cdn.com/v4/letter/f/f05b48/32.png) [@francesco-ballarin](https://fenicsproject.discourse.group/u/francesco-ballarin)\
**Post date:** [June 6, 2024, 12:47pm UTC](https://fenicsproject.discourse.group/t/how-to-define-a-time-dependent-neumann-boundary-condition-if-i-have-known-the-exact-solution/14906/6 "2024-06-06T12:47:50Z")

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Agreed, but that is shown in the tutorial you linked, search for `meshtags`. If I were, I would still try doing it (also) this way, because that will be the way you’ll set Neumann boundary conditions in any practical case (i.e., a case in which you do not have the exact solution, and can ask `ufl` to do `grad(u) times n`).