Numerical Convergence of Discrete Exterior Calculus on Arbitrary Surface Meshes
Removes a long-standing triangulation restriction in DEC, simplifying mesh generation for structure-preserving simulations on surfaces.
The paper shows that discrete exterior calculus (DEC) converges on arbitrary surface meshes, not just Delaunay triangulations, by using a signed diagonal Hodge star. Numerical experiments confirm expected convergence rates for various problems.
Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.