Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions

TL;DR

This paper introduces a systematic framework for decoupling high order elliptic equations into simpler Poisson and Stokes equations using generalized Helmholtz decompositions, enabling superconvergence and broad applicability.

Contribution

It develops a general method for decoupling complex elliptic equations via commutative diagrams and Helmholtz decompositions, applicable to various high order PDEs.

Findings

Decoupling high order elliptic equations into simpler forms.

Achieves superconvergence between Galerkin projection and decoupled approximation.

Provides Helmholtz decompositions for multiple dual spaces.

Abstract

A framework to systematically decouple high order elliptic equations into combination of Poisson-type and Stokes-type equations is developed. The key is to systematically construct the underling commutative diagrams involving the complexes and Helmholtz decompositions in a general way. Discretizing the decoupled formulation leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but not limit to: the primal formulations and mixed formulations of biharmonic equation, fourth order curl equation, and triharmonic equation etc. As a by-product, Helmholtz decompositions for many dual spaces are obtained.