Efficient second-order semi-implicit finite element method for fourth-order nonlinear diffusion equations

TL;DR

This paper introduces a novel second-order mixed finite element method for efficiently solving fourth-order nonlinear diffusion equations by reformulating them as second-order systems, achieving high accuracy and computational efficiency.

Contribution

The paper presents a new second-order fully discrete mixed finite element scheme using backward differentiation and special nonlinear term approximation, reducing computational cost.

Findings

Achieves at least optimal convergence rates matching linear problem estimates.

Demonstrates second-order temporal accuracy through numerical experiments.

Shows the method's efficiency and robustness across various boundary conditions.

Abstract

We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to approximate these equations. In our approach, we propose new techniques using the second-order backward differentiation formula for the time derivative and a special technique for the approximation of nonlinear terms. The use of the proposed technique for nonlinear terms makes the developed numerical scheme efficient in terms of computational cost since the proposed method only deals with a linear system at each time step and no iterative resolution is needed. A numerical convergence study is performed using the method of manufactured and analytical solutions of the system where we investigate different boundary conditions. With respect to the spatial discretization, convergence rates are found to at least match a priori error estimates available for linear problems. The convergence analysis is completed with an investigation of the temporal discretization where we numerically demonstrate the second-order time-accuracy of the proposed scheme using the method of reference solution. We present a series of numerical tests to demonstrate the efficiency and robustness of the proposed scheme.