Postprocessing Mixed Finite Element Methods For Solving Cahn–Hilliard Equation: Methods and Error Analysis

Abstract

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

An Efficient Two-Grid Scheme for the Cahn-Hilliard Equation

“…Therefore this two-grid method can save a lot of computation work compared with standard mixed finite element methods. Indeed, in [38], we have proved that the accuracy of the approximation of our two-grid method is of optimal order. Algorithm 1 Two-grid method 1.…”

A Robust Solver for a Mixed Finite Element Method for the Cahn–Hilliard Equation

“…There is a vast literature on numerical methods for the Cahn-Hilliard equation (cf. [31,12,25,35,37] and the references therein) and solvers based on various numerical schemes were developed in [2,5,9,21,22,23,29,36,28,24,35]. We will consider the mixed finite element method for (1.3)-(1.5) investigated in [11].…”

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

“…The mixed finite element method is very important for solving the partial differential equation. At present, it has been applied to solve elliptic problems [1][2][3][4][5][6], optimal control problems [7,8], plane elasticity problems [9], miscible prob-lems [10], and other problems [11][12][13]. Chen [2,3] developed a new mixed formulation for the numerical solution of second-order elliptic problems.…”