We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate $\Omega$ by a polygonal subdomain $\Omega_h$ and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain $\Omega_h$ and the true domain $\Omega$. Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional order of convergence. A reliable and locally efficient a posteriori error estimator that takes into account the error in the approximation of the boundary data of $\Omega_h$ is also provided.

翻译：我们先验和事后误差分析一种高顺序、可混合的不连续加热金(HDG)方法,该方法适用于在片形弯曲的非多边形域上提出的半线性椭圆问题。我们通过一个多边形子形子体约合$\Omega_h$,并提议一个HDG离散法,根据非线性来源术语的轻度假设和多边形子体子体差价$\Omega_h$和真实域域值$\Omega$之间的距离,该方法被证明是最佳的。此外,还提议并显示一个本地非线性后处理,以提供额外的趋同顺序。还提供了一个可靠和本地高效的事后误差估计器,其中考虑到美元边界数据近似值的错误。