A stabilizer-free $C^0$ weak Galerkin method for the biharmonic equations

TL;DR

This paper introduces a new stabilizer-free weak Galerkin method for biharmonic equations that simplifies implementation while maintaining optimal error estimates, confirmed by numerical experiments.

Contribution

The paper develops a stabilizer-free $C^0$ weak Galerkin method for biharmonic problems, eliminating the need for stabilization terms and simplifying the finite element formulation.

Findings

Achieves optimal error estimates in discrete $H^2$ and $H^1$ norms.

Provides error estimates in $L^2$ norm with optimal or sub-optimal order.

Numerical results confirm theoretical convergence rates.

Abstract

In this article, we present and analyze a stabilizer-free $C^0$ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are $C^0$ continuous piecewise polynomials of degree $k+2$ with $k\geq 0$ and in terms of face unknowns which are discontinuous piecewise polynomials of degree $k+1$. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the $C^1$ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete $H^2$-like norm and the $H^1$ norm for $k\geq 0$ are established for the corresponding WG finite element solutions. Error estimates in the $L^2$ norm are also derived with an optimal order of convergence for $k>0$ and sub-optimal order of convergence for $k=0$. Numerical experiments are shown to confirm the theoretical results.