Elliptic1D Homogenous Dirichlet Boundary Conditions#
Solves the 1D Poisson equation with homogeneous Dirichlet boundary conditions.
\[
-\nabla^2 u(x) = 1
\]
with \(x\in[0,1]\). The boundary conditions are given by
\[
au + b\frac{du}{dx} = g
\]
with
\[
1u(0) + b\frac{du(0)}{dx} = 0
\]
\[
1u(1) + b\frac{du(1)}{dx} = 0
\]
This corresponds to the call to addScalarBC1D of addScalarBC1D(A,b,k,m,dx,dc,nc,v), where dc, nc, and vc are vectors which hold the coefficients for \(a\), \(b\), and \(g\) in the above system of equations. \(a=[1,1]\), \(b=[0,0]\) and \(g=[0,0]\). Substituting these values in gives:
\[
u(0) = 0
\]
\[
u(1) = 0
\]
The true solution is
\[
u(x) = \frac{x(1-x)}{2}
\]
This example is implemented in:
Additional MATLAB/ OCTAVE variants of this example with different boundary conditions: