One dimensional Poisson equation is a known forcing function Boundary condition: 1. Discretizing the domain Dividing the domain into equal segments with the grid-points located at for i = 0, N. The boundaries correspond to i = 0 and i = N.

1. Discretizing the differential equation where and

for N=6, tridiagonal matrix 5 equations for 5 unknowns, it is a closed system and can be solved. But how do it in an efficient way? Let’s rewrite the equation as,

Let’s seek the solution in the form of, Insert Eq.(2) in Eq.(1), Comparing Eq.(2) and Eq.(3), we have, Now we can solve the tridiagonal matrix equation in two steps. First step, we can scan down the leading diagonal starting from N-1 to 1, i=N-1

i=N-2,…,2 i=1 Second step, scanning up the leading diagonal from 0 to N-1 using Eq.(2), i=0 i=1,2,…,N-2, Note that, is the boundary condition.