Recovery based finite element method for biharmonic equation in two dimensional
For researchers in numerical PDEs, this offers a new method for solving fourth-order problems with linear elements, but it is incremental as it builds on existing recovery techniques.
The paper proposes a recovery-based linear finite element method for the biharmonic equation, replacing gradient and Laplace operators with recovery operators and using boundary penalty. Numerical examples on uniform and adaptive meshes demonstrate correctness and effectiveness.
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 $Δ$ 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.