Analysis of a C0 finite element method for the biharmonic problem with Dirichlet boundary conditions

The main focus of this paper is to approximate the biharmonic equation with Dirichlet boundary conditions in a polygonal domain by decomposing it into a system of second-order equations. Subsequently, we explore the regularities exhibited by these equations in each system. Upon demonstrating that the solutions of each resulting system are equivalent to those of the original fourth-order problem in both convex and non-convex polygonal domains, we introduce C0 finite element algorithms designed to solve the decoupled system, accompanied by a comprehensive analysis of error estimates. In contrast to the biharmonic problem, the solutions of the Poisson and Stokes problems display lower regularities, leading to diminished convergence rates for their finite element approximations. This can, in turn, impact the overall convergence rate of the finite element approximation on quasi-uniform meshes for the biharmonic problem. However, we establish an invariant relationship for the source term in the Stokes equation, showing that, under appropriate conditions, the convergence rate of the biharmonic approximation is solely influenced by the Stokes approximation, rather than the first Poisson approximation. To recover the optimal convergence rate for the biharmonic approximation, we also explore the regularities in the weighted Sobolev space and introduce the graded finite element method with the grading parameter only governed by the last Poisson equation. To validate our theoretical insights, we present numerical test results.