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

TL;DR

This paper introduces finite element methods for high-order parabolic equations that ensure mass conservation, positivity, and energy dissipation, with proven convergence and new conservative truncation techniques.

Contribution

The paper develops a novel class of semi-implicit finite element methods that preserve physical properties and introduces new conservative truncation schemes for high-order parabolic equations.

Findings

Methods are proven to converge to established truncation schemes.

Numerical tests demonstrate the efficiency and robustness of the proposed methods.

The approach guarantees mass conservation and positivity in solutions.

Abstract

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 \cite{Berger1975}. 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.