In this paper, a new version of the enriched Galerkin (EG) method for elliptic and parabolic equations is presented and analyzed, which is capable of dealing with a jump condition along a submanifold \({\Gamma _{\text {LG}}}\). The jump condition is known as Henry’s law in a stationary diffusion process. Here, the novel EG finite element method is constructed by enriching the continuous Galerkin finite element space by not only piecewise constants but also with piecewise polynomials with an arbitrary order. In addition, we extend the proposed method to consider new versions of a continuous Galerkin (CG) and a discontinuous Galerkin (DG) finite element method. The presented uniform analyses for CG, DG, and EG account for a spatially and temporally varying diffusion tensor which is also allowed to have a jump at \({\Gamma _{\text {LG}}}\) and gives optimal convergence results. Several numerical experiments verify the presented analyses and illustrate the capability of the proposed methods.

本文提出并分析了椭圆和抛物线方程的丰富Galerkin（EG）方法的新版本，该方法能够处理子流形\（{\ Gamma _ {\ text {LG}}的跳跃条件} \）。跳跃条件在平稳扩散过程中被称为亨利定律。在这里，新颖的EG有限元方法是通过不仅用分段常数而且用任意阶数的分段多项式来丰富连续的Galerkin有限元空间而构造的。此外，我们将提出的方法扩展为考虑连续Galerkin（CG）和不连续Galerkin（DG）有限元方法的新版本。提出的针对CG，DG和EG的统一分析考虑了随时间变化的扩散张量，该张量也允许在\（{\ Gamma _ {\ text {LG}}} \\处跳转，并给出最佳收敛结果。几个数值实验验证了所提出的分析并说明了所提出方法的能力。