Continuous Galerkin and Enriched Galerkin Methods with Arbitrary Order Discontinuous Trial Functions for the Elliptic and Parabolic Problems with Jump Conditions

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 $${\Gamma _{\text {LG}}}$$ Γ 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 $${\Gamma _{\text {LG}}}$$ Γ LG and gives optimal convergence results. Several numerical experiments verify the presented analyses and illustrate the capability of the proposed methods.

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