A stabilizer free weak Galerkin method with implicit $θ$-schemes for fourth order parabolic problems

TL;DR

This work advances the numerical solution of fourth-order parabolic problems by integrating stabilizer-free weak Galerkin methods with implicit $\theta$-schemes in time, achieving stable, high-accuracy full-discrete schemes for $\theta\in[\tfrac12,1]$. It develops semi- and full-discrete formulations based on a weak Laplacian operator, provides comprehensive well-posedness and stability results, and derives sharp error estimates in $H^2$- and $L^2$-type norms, leveraging an elliptic projection $E_h$ for optimal spatial accuracy. Temporal accuracy ranges from first-order (backward Euler) to second-order (Crank–Nicolson) with $\theta=\tfrac12$, and higher-order temporal behavior is detailed for the fully discrete setting. Numerical experiments on triangular and polygonal meshes confirm the predicted convergence rates in space and time, validating the theoretical results and demonstrating robustness across mesh families. The approach offers a streamlined, stabilizer-free alternative to WG methods with solid theoretical guarantees and practical performance for complex geometries.

Abstract

In this paper, we combine the stabilizer free weak Galerkin (SFWG) method and the implicit $θ$-schemes in time for $θ\in [\frac{1}{2},1]$ to solve the fourth-order parabolic problem. In particular, when $θ=1$, the full-discrete scheme is first-order backward Euler and the scheme is second-order Crank Nicolson scheme if $θ=\frac{1}{2}$. Next, we analyze the well-posedness of the schemes and deduce the optimal convergence orders of the error in the $H^2$ and $L^2$ norms. Finally, numerical examples confirm the theoretical results.

Theorems & Definitions (35)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Theorem 3.4
• proof
• Lemma 3.5
• Theorem 3.6
• proof