On a Hybrid Mixed Domain Decomposition Method

TL;DR

This paper introduces a hybridized domain decomposition method inspired by HDG techniques, incorporating stabilization and analyzing its error behavior with promising numerical results.

Contribution

It develops a novel hybrid mixed domain decomposition method with stabilization, providing a variational formulation and error analysis, validated by numerical experiments.

Findings

Convergence rate of q+1 for small stabilization parameter τ.

Discretization error remains bounded in τ.

Observed convergence rates are q+1 for primal and hybrid variables, and q+0.5 for dual variable.

Abstract

We present a domain decomposition formulation based on hybridization which is inspired by hybridized discontinuous Galerkin (HDG) methods, that enhance mixed domain decomposition methods by incorporating stabilization terms. Unlike discontinuous Galerkin methods, our analysis of the proposed finite element method is based on a corresponding consistent variational formulation and a perturbed Galerkin method. In the variational formulation the divergence appears not only within subdomains, but also as an $L^2$-surface quantity on the interfaces. Furthermore, the traces of the finite element functions on the interfaces are replaced by $L^2$-distributions. The well-posedness of the perturbed Galerkin method is shown for an appropriate choice of subspaces, in a manner similar to that of the variational formulation. For the finite element method we use Raviart-Thomas elements for the dual variable and piecewise polynomials for the primal and hybrid variables, respectively. We perform an analysis of the discretization error which is explicit in the stabilization parameter $\tau$. Numerical experiments for piecewise smooth solutions using finite element spaces of order~$q$ on curved quadrilateral meshes confirm the predicted convergence rate of $q+1$ for small values of $\tau$. In the error analysis we observe the discretization error to be uniformly bounded in $\tau$. Even for large $\tau$ values the observed convergence rates for the primal and for the hybrid variables are $q+1$. For the dual variable the convergence rate depends on the stabilization parameter and the mesh-width, with an asymptotic rate of $q+\tfrac12$.