Neumann Boundary Condition in FEM
I am trying to implement a small FEM code for solving Poisson equation on arbitrary domains. I understand how to implement Drichlet Boundary conditions after formulating the Global stiffness matrix.
But I do not quite understand how to implement Neumann Boundary conditions. For example I have to integrate over the the boundary of the element. But what if a particular element does not have a boundary, like its completely in the interior.
Also when we integrate over the boundary what should be done when a element has only one of its nodes on the boundary ?
Thanks in advance
The Poisson equation u,ii=f FEM formulation is derived using the usual Galerkin shape-function-weighted integration and the Gauss divergence theorem. You get the element boundary term as the surface integral over the element of (NI du/dn), where NI is the shape function of node I and du/dn is the derivative of u in the the outward-normal direction to the face.
When the element is internal, the same face is shared by two elements, so this term cancels in the assembly. This is also true if a single node of it is on the boundary, but the face is internal.
For en external face, Neumann BC means the du/dn is prescribed, so the integral is readily calculated. In the special case du/dn=0 (no flux), this term is 0.
I hope this helps,
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