Elliptic2D X Dirichlet Y Dirichlet

Solves the 2D Poisson equation with Dirichlet boundary conditions

\[- \nabla^2 u(x,y) = 0\]

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

The boundary conditions are given by

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

with $a = 1$, $b = 0$, and $g(x,y) = e^x \cos y$, which is equivalent to Dirichlet conditions along each boundary. This corresponds 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 $= (1.0, 1.0, 1.0, 1.0)$, nc $= (0.0, 0.0, 0.0, 0.0)$, and v is composed of 4 vectors: $g$ along the left boundary, $g$ along the right boundary, $g$ along the bottom boundary, and $g$ along the top boundary.

The true solution is

\[u(x,y) = e^x \cos y\]

This example is implemented in elliptic2DXDirichletYDirichlet.jl