Recovery based finite element method for biharmonic equation in two dimensional

TL;DR

This paper introduces a recovery-based linear finite element method for solving the biharmonic equation in two dimensions, utilizing a recovery operator to enhance the approximation of derivatives and employing boundary penalty for conditions.

Contribution

The paper proposes a novel recovery-based finite element approach for biharmonic equations, replacing derivatives with a recovery operator and incorporating boundary penalty techniques.

Findings

Method is validated through numerical examples on uniform and adaptive meshes.

The approach effectively solves the biharmonic equation with demonstrated accuracy.

Explicit matrix formulation facilitates implementation and analysis.

Abstract

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $\Delta$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.