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 $τ$. 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 $τ$. In the error analysis we observe the discretization error to be uniformly bounded in $τ$. Even for large $τ$ 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 $q+\tfrac12$.

翻译：我们提出一种基于杂交的域分解形式，该方法受混合间断伽辽金格式启发，通过引入稳定化项改进混合域分解方法。与间断伽辽金方法不同，我们对所提有限元方法的分析基于相应的相容变分提法和摄动伽辽金方法。在变分提法中，散度不仅出现在子域内部，还以$L^2$曲面量的形式出现在界面上。此外，有限元函数在界面上的迹被替换为$L^2$分布。通过类似于变分提法的子空间恰当选择，证明了摄动伽辽金方法的适定性。对于有限元方法，我们分别采用Raviart-Thomas元处理对偶变量，以及分片多项式处理原始变量和混合变量。我们对离散误差进行了显式依赖于稳定化参数$τ$的分析。对于曲边四边形网格上阶数为$q$的有限元空间，针对分片光滑解的数值实验验证了当$τ$较小时$q+1$阶的预测收敛速率。在误差分析中，我们发现离散误差关于$τ$一致有界。即使对于大$τ$值，原始变量和混合变量的观测收敛速率仍为$q+1$。对于对偶变量，收敛速率依赖于稳定化参数和网格宽度，其渐近收敛速率为$q+\tfrac12$。