Special Session 39: Recent Developments in Gradient Flows: Theory, Numerics, and Applications
Super-convergent HDG methods for the Cahn-Hilliard equation
Daozhi Han
State University of New York at Buffalo USA
Co-Author(s): Gang Chen, Jiaxuan Liu, John Singler, Yangwen Zhang, Dujin Zuo
Abstract:
We present and analyze a super-convergent hybridizable discontinuous Galerkin method for solving the Cahn-Hilliard equation. The HDG method utilizes polynomials of degree k+1 for scalar variables and polynomials of degree k for fluxes and numerical traces with reduced diffusion stabilization. For the classical Cahn-Hilliard equation we establish optimal convergence rates for all variables and all polynomial orders with error constants depending on the inverse of interface thickness in polynomial orders. For the advective Cahn-Hilliard equation we show that optimal convergence can be obtained without any advection stabilization. The key tools involved include the HDG spectral estimate of the linearized Cahn-Hilliard operator and new elliptic projection adapted to advection.