Poisson's Equation on the US Map

The region is specified by a description of the boundary, called a skeleton . The skeleton also describes any internal interfaces, in this case state lines.

This initial mesh was created from the skeleton. It is colored by element diameter. All internal interfaces are incorporated into the mesh as element edges. On the right is the same mesh, but without the triangle boundaries drawn in black.

We solved Poisson's equation (-Laplace u = 1) in this domain, with homogeneous Dirichlet boundary conditions. On the left is the final mesh with 100,000 vertices, created using adaptive local mesh refinement. On the right is the final a posteriori error estimate for this problem.

Here are some perspective plots of the solution.