On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

Abstract

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.