In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method.

在这项工作中，我们介绍了丰富的Galerkin（EG）方法的一般化。我们方案的关键特征是自适应两网格方法，除了通过不连续的自由度对标准有限元离散化进行标准富集之外，还允许以非一致性方式细分选定的（例如，有问题的）网格单元，并且在这个更细的网格上使用进一步的不连续浓缩。我们证明稳定性，并先验先验通过使用专门定制的投影并对两个网格进行标准收敛分析的某些部分，可以估算线性对流方程的误差。通过在粗网格和细网格上都允许任意程度的富集（也包括不富集的情况），我们的分析技术非常笼统，因为我们的结果涵盖了从标准连续有限元方法到网格的范围。标准的不连续伽勒金（DG）方法，带有（或不带有）局部亚细胞富集。数值实验证实了我们的分析结果，并表明了该方法的良好鲁棒性。