How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

Question

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet boundary condition but I haven't found any mathematical justification for this. It seems to me that setting essential boundary conditions should probably minimize some functional of the error (e.g. minimize $ ||u -u_h|| $ over the portion of the boundary that the Dirichlet BC is applied ) even though this would be more computationally expensive.

Is there any justification for setting the BC like this and if so, what would the proper norm be?

Answer 1

There is mathematical justification for setting Dirichlet boundary degrees of freedom to a value. However, you should adjust your variational form accordingly. If you are looking at a general problem, say:

Find $u\in\mathcal{U}$ such that

$a(u,w)=l(w) \ \ \forall w\in\mathcal{V}$

where

$\mathcal{U}=\{u:\int \nabla u^2 < \infty, u=g\text{ on }\Gamma_D\}$

$\mathcal{V}=\{u:\int \nabla u^2 < \infty, u=0\text{ on }\Gamma_D\}$

Instead we can write $u = v + g$ where $v\in\mathcal{V}$ and $g$ is the Dirichlet condition. Then the variational form becomes

$a(v+g,w)=l(w)$

or by using the linearity of $a(.,.)$

$a(v,w)=l(w)-a(g,w)$

In a finite element code, you can form your element stiffness matrix as if there were no boundary conditions. Then you take the column of the local matrix which corresponds to the Dirichlet boundary condition, scale it by the coefficient you want to enforce, and subtract it from the right-hand-side. This is the discrete form of what I wrote above, $-a(g,w)$. Then you zero out that column and the corresponding Dirichlet row, placing a 1 in the diagonal and the coefficient you wish to enforce. This decouples the equation from the system and yet sets the value you wish to enforce.

I recommend The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, by Tom Hughes. He has an expanded discussion of this issue starting on page 8.

Answer 2

To add up to Nathan's great answer with the variational reasoning, one often needs algorithmic details when implementing finite elements. For example,

I also have a more detailed explanation on the subject in my personal notes. Please see the chapter "Constrained Linear Systems".