Mass-conservative and positivity preserving second-order semi-implicit methods for high-order parabolic equations

We consider a class of finite element approximations for fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. In our approach, we first solve a variational problem and then an optimization problem to satisfy the desired physical properties of the solution such as conservation of mass, positivity (non-negativity) of solution and dissipation of energy. Furthermore, we show existence and uniqueness of the solution to the optimization problem and we prove that the methods converge to the truncation schemes [10]. We also propose new conservative truncation methods for high-order parabolic equations. A numerical convergence study is performed and a series of numerical tests are presented to show and compare the efficiency and robustness of the different schemes. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study introduces a new finite element approximation method for fourth-order parabolic equations, along with a numerical convergence study and tests demonstrating the effectiveness and robustness of the method. The results indicate that the proposed method can satisfy the desired physical properties of the solution and converge to the truncation schemes.