Auto-Stabilized Weak Galerkin Finite Element Methods for Biharmonic Equations on Polytopal Meshes without Convexity Assumptions

Abstract: This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for biharmonic equations with built-in stabilizers. Unlike existing stabilizer-free WG methods limited to convex elements in finite element partitions, our approach accommodates both convex and non-convex polytopal meshes, offering enhanced versatility. It employs bubble functions without the restrictive conditions required by existing stabilizer-free WG methods, thereby simplifying implementation and broadening application to various partial differential equations (PDEs). Additionally, our method supports flexible polynomial degrees in discretization and is applicable in any dimension, unlike existing stabilizer-free WG methods that are confined to specific polynomial degree combinations and 2D or 3D settings. We demonstrate optimal order error estimates for WG approximations in both a discrete $H^2$ norm for $k\geq 2$ and a $L^2$ norm for $k>2$, as well as a sub-optimal error estimate in $L^2$ when $k=2$, where $k\geq 2$ denotes the degree of polynomials in the approximation.