Decoupled finite element methods for a fourth-order exterior differential equation
Xuewei Cui, Xuehai Huang
TL;DR
The paper addresses a fourth-order exterior differential equation on a bounded domain and introduces decoupled finite element methods based on Helmholtz (Hodge) decomposition and finite element exterior calculus. It decomposes the problem into two second-order exterior equations and a generalized Stokes system, and presents both quotient-space and quotient-free decoupled variational formulations, with the latter avoiding quotient spaces for easier discretization. A conforming finite element framework is developed, featuring spaces for $\phi$, $p$, $r$, $u$, and $w$ that lead to a MINI-like Stokes discretization and allow local elimination of auxiliary variables to yield SPD systems; rigorous error analysis demonstrates optimal convergence rates in key norms, and numerical experiments in three dimensions verify the theoretical results for the biharmonic case. The work advances practical, stable solvers for high-order exterior differential equations by avoiding quotient spaces and leveraging discrete de Rham complexes.
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.