In this work we propose and analyze an HDG method for the Stokes equation whose domain is discretized by two independent polygonal subdomains with different meshsizes. This causes a non-conformity at the intersection of the subdomains or leaves a gap (unmeshed region) between them. In order to appropriately couple these two different discretizations, we propose suitable transmission conditions to preserve the high order convergence of the scheme. Furthermore, stability estimates are established in order to show the well-posedness of the method and the error estimates. In particular, for smooth enough solutions, the $L^2$ norm of the errors associated to the approximations of the velocity gradient, the velocity and the pressure are of order $h^{k+1}$, where $h$ is the meshsize and $k$ is the polynomial degree of the local approximation spaces. Moreover, the method presents superconvergence of the velocity trace and the divergence-free postprocessed velocity. Finally, we show numerical experiments that validate our theory and the capacities of the method.

在这项工作中，我们提出并分析了Stokes 方程的 HDG 方法，其域由两个具有不同网格尺寸的独立多边形子域离散化。这会导致子域交叉处的不一致或在它们之间留下间隙（未划分网格的区域）。为了适当地耦合这两种不同的离散化，我们提出了合适的传输条件来保持方案的高阶收敛。此外，建立稳定性估计是为了显示该方法的适定性和误差估计。特别是，对于足够平滑的解，${\mathrm{大号}}^2$与速度梯度近似相关的误差范数，速度和压力是有序的$H^{ķ+1}$， 在哪里$H$是网格大小和$ķ$是局部逼近空间的多项式次数。此外，该方法呈现了速度轨迹的超收敛性和无散度后处理速度。最后，我们展示了验证我们的理论和方法能力的数值实验。