Elliptic 2D X Periodic Y Dirichlet

Solves the 2D Poisson equation with periodic boundary conditions in the x-direction and Dirichlet boundary conditions in the y-direction

\[- \nabla^2 u(x,y) = 2 \sin (2 \pi x) (1 + 2 \pi^2 y (1 - y))\]

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

The bottom and top boundary conditions are given by

\[a u + b \nabla u \cdot \hat n = g\]

with $a = 1$, $b = 0$, and $g(x, y) = y (1 - y) \sin (2 \pi x)$, which is equivalen to Dirichlet boundary conditions along the two boundaries. This correspondes to the call to addScalarBC! of addScalarBC!(A, b, k, m, dx, n, dy, bc), where A is the operator matrix, b is the right hand side vector, and bc is a boundary condition struct that contains the tuples dc, nc, and v which hold the coefficients for $a$, $b$, and $g$ in the above systems of equations. dc = $= (0.0, 0.0, 1.0, 1.0)$, nc $= (0.0, 0.0, 0.0, 0.0)$, and v is composed of 4 elements: the first two are never accessed because of the periodicity in the x-direction, the third is $g$ along the bottom boundary, and the fourth is $g$ along the top boundary.

The true solution is

\[u(x,y) = y (1 - y) \sin (2 \pi x)\]

This example is implemented in elliptic2DXPerYDirichlet.jl