Enriched Galerkin finite element methods for the first order hyperbolic problems

초록

In this paper, we introduce general enriched Galerkin finite element spaces, the conforming finite elements of degree k, with the enrichment of element-wise ℓ degree polynomials. This general enriched finite element is employed to solve the linear hyperbolic equation. The construction of the bilinear form for the hyperbolic equation is based on that of the elegant Brezzi–Marini–Süli jump stabilized discontinuous Galerkin finite elements. We demonstrate the proposed enriched Galerkin finite element can lead to the optimal error estimates for any degree k if the degree of enrichment ℓ is at least k−1 with ℓ=0 for k=0. This shows that the standard enriched Galerkin finite element method, which takes ℓ=0 for general k, leads to optimal order error only for k=0 or k=1. However, the standard enriched Galerkin finite element methods may not achieve the optimal order accuracy for k≥2. We justify our theoretical findings by sample numerical experiments. © 2026 .

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