  We introduce a framework for solving a class of parabolic partial
differential equations on triangle mesh surfaces, including the Hamilton-Jacobi
equation and the Fokker-Planck equation. PDE in this class often have nonlinear
or stiff terms that cannot be resolved with standard methods on curved triangle
meshes. To address this challenge, we leverage a splitting integrator combined
with a convex optimization step to solve these PDE. Our machinery can be used
to compute entropic approximation of optimal transport distances on geometric
domains, overcoming the numerical limitations of the state-of-the-art method.
In addition, we demonstrate the versatility of our method on a number of linear
and nonlinear PDE that appear in diffusion and front propagation tasks in
geometry processing.
