Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions
For researchers in numerical PDEs, this provides a systematic decoupling method for high-order elliptic equations, but the results are theoretical and incremental over existing Helmholtz decomposition techniques.
The paper develops a framework to decouple high-order elliptic equations into Poisson- and Stokes-type equations using generalized Helmholtz decompositions, achieving natural superconvergence between Galerkin projection and decoupled approximation. Examples include biharmonic, fourth-order curl, and triharmonic equations.
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.