The stabilizer-free weak Galerkin finite element method for the Biharmonic equation using polynomials of reduced order

TL;DR

This work develops a stabilizer-free weak Galerkin method for the biharmonic equation by employing reduced-order polynomials on the boundary and a modified weak Laplacian. The method builds discrete spaces $V_h$ with interior and boundary components and defines a discrete weak Laplacian $\Delta_w$, enabling a stabilizer-free formulation that yields optimal $H^2$ and $L^2$ error estimates. The authors prove well-posedness, derive error equations, and establish convergence rates of $O(h^{k-1})$ in $H^2$ and $O(h^{k+k_0-2})$ in $L^2$ (with $k_0=\min\{k,3\}$), corroborated by numerical experiments on triangular and polygonal meshes. The results demonstrate efficient DOF reduction without stabilizers and provide a viable pathway for high-order discretizations of fourth-order problems. These findings have potential impact on practical simulations requiring accurate biharmonic solvers with simplified stabilization and reduced boundary data handling.

Abstract

In this article, we decrease the degree of the polynomials on the boundary of the weak functions and modify the definition of the weak laplacian which are introduced in \cite{BiharmonicSFWG} to use the SFWG method for the biharmonic equation. Then we propose the relevant numerical format and obtain the optimal order of error estimates in $H^2$ and $L^2$ norms. Finally, we confirm the estimates using numerical experiments.

Theorems & Definitions (23)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Theorem 3.1