A staggered cell-centered DG method for the biharmonic Steklov problem on polygonal meshes: A priori and a posteriori analysis

In this paper, a staggered cell-centered discontinuous Galerkin method is developed for the biharmonic problem with the Steklov boundary condition. Our approach utilizes a first-order system form of the biharmonic problem and can handle fairly general meshes possibly including hanging nodes, which favors adaptive mesh refinement. Optimal order error estimates in 𝐿² norm can be proved for all the variables. Moreover, the approximation of the primal variable superconverges in 𝐿² norm to a suitably chosen projection without requiring additional regularity. Residual type error estimators are proposed, which can guide adaptive mesh refinement to deliver optimal convergence rates even for solutions with singularity. Numerical experiments confirm that the optimal convergence rates in 𝐿² norm can be achieved for all the variables. Moreover, all the provided residual type error estimators show the desired results. In particular, the numerical results demonstrate that the proposed scheme on a polygonal approximation of the disk works well for the classic Babuška example.