Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in -norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in -norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.