Enforcing Dirichlet Conditions

For the Dirichlet Problem, the boundary values are prescribed at the boundary degrees of freedom $B$. Let $I$ denote the interior DOFs. Partition the system $L\mathbf{u}=\mathbf{b}$:

$$\left(\begin{array}{cc}{L}_{II}& L_{IB}\\ L_{BI}& L_{BB}\end{array}\right)\left(\begin{array}{c}{\mathbf{u}}_I\\ {\mathbf{u}}_B\end{array}\right)=\left(\begin{array}{c}{\mathbf{b}}_I\\ {\mathbf{b}}_B\end{array}\right),{\mathbf{u}}_B=\mathbf{f}.$$

Elimination (symmetric). Solve $L_{II}{\mathbf{u}}_I={\mathbf{b}}_I-L_{IB}\mathbf{f}$. The reduced matrix $L_{II}$ is symmetric positive definite, so CG applies.

Row/column replacement (same size). Keep the full system but, for each $i\in B$, zero out row $i$ and column $i$, set the diagonal to $1$, set $b_i=f_i$, and move the eliminated column contributions $-L_{IB}\mathbf{f}$ to the right-hand side. This keeps the matrix size and symmetry and is equivalent to elimination. Zeroing only the rows is simpler, but it destroys symmetry.

Penalty / Nitsche. Add $\frac{\kappa }{h}{\int }_{\partial \Omega }(u-f)v$ with a large penalty. This is approximate unless the symmetric Nitsche terms are included.

Measured quantity. In the Dirichlet setting one measures the boundary current $g=\gamma {\partial }_{\nu }u$. The consistent discrete way to obtain it is the residual of the unconstrained boundary rows, ${\mathbf{r}}_B=(L\mathbf{u}{)}_B$. By the weak form, $(r_B{)}_i\approx {\int }_{\partial \Omega }g{\phi }_i\mathrm{d}s$, so the nodal current density follows from $M_{\Gamma }\mathbf{g}={\mathbf{r}}_B$ (see Boundary Mass and Stiffness Matrices). This variational flux recovery is more accurate than differentiating $u_h$ numerically.

References

1. A. Ern, J.-L. Guermond (2004). Theory and Practice of Finite Elements. Springer. doi:10.1007/978-1-4757-4355-5
2. S. C. Brenner, L. R. Scott (2008). The Mathematical Theory of Finite Element Methods, 3rd ed. Springer. doi:10.1007/978-0-387-75934-0