Decoupled finite element methods for a fourth-order exterior differential equation

TL;DR

This paper introduces decoupled finite element methods for solving a fourth-order exterior differential equation, simplifying the problem by decomposing it into lower-order equations and avoiding quotient spaces, with numerical validation in 3D.

Contribution

It presents a novel decoupled finite element approach based on differential complexes and Helmholtz decomposition for fourth-order exterior differential equations.

Findings

Decoupled methods successfully solve 3D biharmonic equations.

Avoids the use of quotient spaces in finite element formulations.

Numerical results verify the effectiveness of the proposed methods.

Abstract

This paper proposes novel decoupled finite element methods for a fourth-order exterior differential equation. Based on differential complexes and the Helmholtz decomposition, the fourth-order exterior differential equation is decomposed into two second-order exterior differential equations and one generalized Stokes equation. A key advantage of this decoupled formulation is that it avoids the use of quotient spaces. A family of conforming finite element methods are developed for the decoupled formulation. Numerical results are provided for verifying the decoupled finite element methods of the biharmonic equation in three dimensions.