Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions
Abstract
A framework to systematically decouple high order elliptic equations into a 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 are not limited to the primal formulations and mixed formulations of the biharmonic equation, fourth order curl equation, and triharmonic equation. As a byproduct, Helmholtz decompositions for many dual spaces are obtained.
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