Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source

Two fully discrete, discontinuous Galerkin schemes with time-dynamic, locally refined meshes in space are developed for a fourth-order Cahn–Hilliard equation with an added nonlinear reaction term, a phenomenological model that can describe cancerous tumour growth. The proposed schemes, which are both second-order in time, are based on a primitive-variable discontinuous Galerkin spatial formulation that is valid in any number of space dimensions. We prove that the schemes are convergent, with optimal-order error bounds, even in the case where the mesh is changing with time, provided that the number of mesh changes is bounded by some constant. The schemes represent flexible, high-order accurate alternatives to the standard mixed C⁰ finite element methods and nonconforming (plate) finite element methods for solving fourth-order parabolic partial differential equations. We conclude the paper with tests showing the convergence of the scheme at the predicted rates and the flexibility of the method for approximating complex solution dynamics efficiently.