Elliptic1D Pure Robin Boundary Conditions

Elliptic1D Pure Robin Boundary Conditions#

Solves the 1D Poisson boundary value problem with pure Robin boundary conditions.

\[ -\nabla^2 u(x) = \pi^2 \sin(\pi x) \]

with $x\in[0,1]$.

The boundary conditions are given by

\[ au + b\frac{du}{dx} = g \]

with the left hand side boundary condition (Robin) satisfying

\[ -200u(0) + 1\frac{du(0)}{dx} = 10 \]

and the right hand boundary condition (Robin) satisfying $$ 400u(1) + 1\frac{du(1)}{dx} = 15 $$

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. $a=[-200,400]$, $b=[1,1]$ and $g=[10,15]$. Substituting these values in gives:

\[ -200u(0) + \frac{du(0)}{dx} = 10 \]

\[ 400u(1) + \frac{du(1)}{dx} = 15 \]

The exact solution is:

\[ u(x) = \sin(\pi x) + \frac{35 - \pi}{403}x + \frac{402\pi - 3995}{80600} \]

The example is taken from this paper

This example is implemented in:

MATLAB/ OCTAVE

Additional MATLAB/ OCTAVE variants of this example with different boundary conditions: