How to incorporate boundary conditions into a finite element iterative solver without plugging them in directly?

Question

I'm looking for papers on solving finite element problems without plugging in the boundary values directly, but I'm not having great luck.

Specifically, I want to leave the boundary values of my finite element system as unknowns (say the interior values are $x_0$ and the boundary values are $x_1$) and then form the linear system

$$ \left[\matrix{A_0 \\ A_1}\right]\left[\matrix{x_0 \\ x_1}\right] = \left[\matrix{f_0 \\ f_1}\right]$$

where $f_0$ and $f_1$ are known, and then when I use a linear system solver (jacobi, GS, CG, etc), I'd like to impose the boundary conditions as part of the iteration.

Is there a simple way to do this?

I think that this will be advantageous since it will preserve some of the symmetry from the structure of the mesh.

Answer 1

You can always write the system as

$$\begin{pmatrix}A&B\\0&I\end{pmatrix}\begin{pmatrix}x_{interior}\\x_{boundary}\end{pmatrix}=\begin{pmatrix}f\\0\end{pmatrix},$$ where $I$ is the identity matrix.

Answer 2

Maybe you could try Nitsche's method? This results in a SPD matrix.