Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes without Convexity Constraints
<|reference_start|>Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes without Convexity Constraints: This paper introduces an auto-stabilized weak Galerkin (WG) finite element method with a built-in stabilizer for Poisson equations. By utilizing bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, marking a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches and is applicable in any dimension $d$, offering substantial advantages. The proposed method maintains a simple, symmetric, and positive definite structure. These benefits are evidenced by optimal order error estimates in both discrete $H^1$ and $L^2$ norms, highlighting the effectiveness and accuracy of our WG method for practical applications.<|reference_end|>
@article{wang2024auto-stabilized,
    title={Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes
  without Convexity Constraints},
    author={Chunmei Wang},
    journal={arXiv preprint arXiv:2408.11927},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.11927},
    primaryClass={math.NA cs.NA}
}