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.

我们提出了应用于分段弯曲非多边形域上的半线性椭圆问题的高阶可混合不连续伽辽金 (HDG) 方法的先验和后验误差分析。我们通过多边形子域\(\Omega _h\)近似\(\Omega \)并提出 HDG 离散化，这在与非线性源项和边界之间的距离相关的温和假设下被证明是最佳的多边形子域\(\Omega _h\)和真实域\(\Omega \)。此外，还提出并显示了标量未知数的局部非线性后处理，以提供额外的收敛顺序。一个可靠且本地高效的还提供了后验误差估计器，该估计器考虑了\(\Omega _h\)的边界数据的近似误差。