How to solve [K]{U}={F} FEM where some values of {U} are known.

Question

I am trying to solve the 2D-Poisson equation using the FEM. I have created the global stiffness matrix and I know my boundary conditions. I am looking at $0<x<4$ and $0<y<2$ with boundary conditions U(0,y) = 100, U(4,y) = 250, U(x,0) = 50, U(x,2) = 200. I am using square Elements (8 of them) broken at each integer. So i know all the U-values except for U6,U7,U8 (the values inside the domain, not on the boundary). How do I solve for these? Also, since it is the poisson equation {F} = {0.0}. Hopefully it will help to note that there are 15 values of U, 12 of which are known due to boundary conditions.

Answer 1

Adapted from this lecture notes page 9.

The approach splits the matrix and right-hand side into Dirichlet and free nodes: $$ K = \begin{pmatrix} K_{DD} & K_{Df} \\ K_{fD} & K_{ff} \end{pmatrix} \qquad F = \begin{pmatrix} F_D \\ F_f \end{pmatrix} $$

In your case the free nodes have the indices $\{6, 7, 8\}$. With $N$ the total number of nodes ($N = 15$ in your case) the system of linear equations for $U = (U_i) \in \mathbb{R}^N, U=(U_D, U_f)$ is then: $$ \begin{pmatrix} I & 0 \\ K_{fD} & K_{ff} \end{pmatrix} \begin{pmatrix} U_D \\ U_f \end{pmatrix} = \begin{pmatrix} U_D \\ F_f \end{pmatrix} $$

From the second row of the equation you get: $$K_{fD} U_D + K_{ff} U_f = F_f$$

Bringing the unknowns on one side and $F_f = 0$: $$U_f = - K_{ff}^{-1} K_{fD} U_D$$