C\(^0\)-Interior Penalty Discontinuous Galerkin Approximation of a Sixth-Order Cahn-Hilliard Equation Modeling Microemulsification Processes
Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C\(^0\) Interior Penalty Discontinuous Galerkin (C\(^0\)IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C\(^0\) conforming Lagrangian finite elements of polynomial degree \(p \ge 2\). For the semidiscretized problem we derive quasi-optimal a priori error estimates for the global discretization error in a mesh-dependent C\(^0\)IPDG norm. The semidiscretized problem represents an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order \( q \ge 2\). Numerical results show the formation of microemulsions in an oil/water system and confirm the theoretically derived convergence rates.
Ronald H. W. Hoppe acknowledges support by the NSF grants DMS-1115658, DMS-1216857, DMS-1520886 and by the German National Science Foundation DFG within the Priority Program SPP 1506.
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