Published

Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions

Long Chen and Xuehai Huang

SIAM Journal on Numerical Analysis, 56(5), 2796-2825, 2018

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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.