# Non-linear stationary heat quation in 3D

**URL:** <https://fenicsproject.discourse.group/t/non-linear-stationary-heat-quation-in-3d/4378>\
**Category:** variational formulation\
**Created:** [October 22, 2020, 4:09pm UTC](https://fenicsproject.discourse.group/t/non-linear-stationary-heat-quation-in-3d/4378 "2020-10-22T16:09:08Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![VKTR](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/vktr/32/2119_2.png) [@VKTR](https://fenicsproject.discourse.group/u/VKTR)\
**Post date:** [October 22, 2020, 4:09pm UTC](https://fenicsproject.discourse.group/t/non-linear-stationary-heat-quation-in-3d/4378/1 "2020-10-22T16:09:08Z")

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Hi there,

I’m trying to solve the heat equation in 3D without Dirichlet-BC and two Neumann-BC’s. One represents a spatially constant heat inflow that is applied to a subdomain of the boundary. The other non-linear term represents the heat radiation.

However the solver does not converge. I’m not too familiar with non-linear PDE’s, so my question is how to look for the problem.

Here you see a minimum example:

```
from fenics import *

def bottom(x, on_boundary):

    return x[2]<-.99 and on_boundary

def top(x, on_boundary):

    return x[2]> .99 and on_boundary

class Top(SubDomain):

    def __init__ (self):

        super(). __init__ ()

    def inside(self,x, on_boundary): 

        return top(x, on_boundary)

# define mesh and functionspace

mesh = BoxMesh(Point(-1,-1,-1), Point(1,1,1),10,10,10)

V = FunctionSpace(mesh, 'CG', 1)

# mark subdomain

sub_domains = MeshFunction('size_t', mesh, mesh.topology().dim()-1)

sub_domains.set_all(0)

top_domain = Top()

top_domain.mark(sub_domains, 1)

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

# define and solve variational formulation

u0 = Constant(293.15)

u = Function(V)

v = TestFunction(V)

F = inner(grad(u) ,grad(v)) * dx + (u **4-u0** 4)*v*ds - v*ds(1)       

solve(F == 0, u, solver_parameters={"newton_solver":{"relative_tolerance": 1e-6}})
```

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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:** [October 22, 2020, 4:33pm UTC](https://fenicsproject.discourse.group/t/non-linear-stationary-heat-quation-in-3d/4378/2 "2020-10-22T16:33:55Z")

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Modify your initial guess to be closer to your wanted temperature. Use for instance:

```auto
u.vector()[:] = 200

```

before you solve and the newton solver will converge

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<div class="post-metadata">

**Author:** ![VKTR](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/vktr/32/2119_2.png) [@VKTR](https://fenicsproject.discourse.group/u/VKTR)\
**Post date:** [October 23, 2020, 7:52am UTC](https://fenicsproject.discourse.group/t/non-linear-stationary-heat-quation-in-3d/4378/3 "2020-10-23T07:52:36Z")

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Now it works, thanks!