Numerical analysis of a second-order energy-stable finite element method for the Swift-Hohenberg equation

Abstract

In this work, we first prove the existence, uniqueness and regularity of the solution of the Swift-Hohenberg equation by applying the Galerkin spectral method. Then we investigate the convergence of a finite element method in the mixed formulation for the Swift-Hohenberg equation, with Crank-Nicolson scheme in time discretization. We prove that our semidiscrete and fully discrete numerical schemes satisfy unique solvability and unconditional energy stability. Moreover, we prove optimal error estimates for the schemes. Finally, numerical tests are given to validate our theoretical results.