Default boundary conditions for this heat equation

mathematics
1

Hi. Assume I am solving a multidimensional heat equation in FeniCS where

\frac{\partial \Upsilon}{\partial t}(t,x,y,z) = \nabla^2\Upsilon(t,x,y,z)

with

x\in[x_0,x_1], \\ y\in[y_0,y_1], \\ z\in[z_0,z_1]

on a unit cube. If I don’t prescribe any Dirichlet boundary conditions and simply provide an initial condition \Upsilon(t=0,x,y,z) = \Upsilon_{0}, will FeniCS prescribe (natural) Neumann boundary conditions of the form

\partial_{x}\Upsilon(t,x_{0},y,z) = \partial_x\Upsilon(t,x_1,y,z) = 0,\\ \partial_{y}\Upsilon(t,x,y_0,z) = \partial_y\Upsilon(t,x,y_1,z) = 0,\\ \partial_{z}\Upsilon(t,x,y,z_0) = \partial_z\Upsilon(t,x,y,z_0) = 0

?

2

Yes. This is because fenics works with the variational form of the problem, and thus will prescribe an homogenuous boundary condition resulting from integration by parts.

In this problem, you only do integration by parts of the laplacian, resulting in the sketched boundary condition. For other problems, such as the Stokes problem, you will get different natural boundary conditions in FEniCS depending on how you treat your variational form.

1 Like
3

Thank you Dr. Dokken.