*have*covered, see posts with the tag

*finite element methods*. These include, e.g., basic diffusion in planar domains, wave equations in planar domains using symplectic methods, and electromagnetic wave simulation using vector finite elements. These posts will be a useful preview of today's post.

Our first two examples concern heat diffusion on evolving surfaces with two interesting topologies: the sphere and the torus. The evolution is fairly simple: rotation about an axis. The heat source remains fixed in the surrounding 3D space, and focuses on a fixed (in the ambient space) spot on the sphere; the rotation carries the heat away and it spreads throughout the sphere (or torus).

Next, we consider a case where the solution can't be effected by an obvious or simple change of coordinates (i.e. in the terminology of fluids and continuum mechanics, switching from Eulerian to Lagarangian coordinates): heat on a bouncing sphere that stretches and shrinks vertically. Here the heat source builds with time and cuts off midway to just let the heat spread:

The implementation of all of these cases are via surface finite elements in the Eulerian description: the surface is considered a material that moves through space, but the spatial coordinates are fixed.

Our last example is a wave equation on a sphere, using surface finite elements and symplectic methods. The oscillating quantity $u$ is depicted as displacement along the normal direction of a sphere (so that, for example, if $u$ has some small positive value at one point and decreases rapidly to 0 as you move away, it looks like a small lump over that point). The initial condition is a ruffly structure (with speed 0), and we just let it go.

Enjoy!