A novel approach of superconvergence analysis of the bilinear-constant scheme for time-dependent Stokes equations
The paper considers the superconvergence of a low-order fully-discretized time-dependent Stokes discretization in two space dimensions. In the introduction, the time-dependent Stokes problem is discussed with its importance as a stepping stone for the Navier-Stokes equations. The author mentions various publications on different discretization techniques for the non-transient Stokes equations, including superconvergence results for a bilinear-constant finite element pair on two-dimensional rectangular meshes. The main result of this paper is to extend this result to the time-dependent Stokes equations discretized by a backward Euler time discretization. The second chapter states some preliminaries detailing the assumptions on mesh and ansatz functions as well as introducing a crucial interpolation operators for velocity and pressure. Furthermore, well-known interpolation results for a Stokes projection operator are stated. Finally, the results for the interpolation operators originally proven by \textit{J. Pan} [SIAM J. Numer. Anal. 34, No. 6, 2424--2430 (1997; Zbl 0894.76045)] are formulated in Lemma 2.1. With these preliminaries, the author proves in Lemma 2.2 error bounds between the Stokes projection operator and the interpolation operator making use of a discrete inf-sup condition. Lemma 2.3 then introduces a discrete Gronwall inequality, and finally the fully discretized weak form (using a backward Euler scheme) is presented. The third section is concerned with the paper's main result, superconvergence for the proposed scheme. Theorem 3.1. provides error bounds for the fully discretized scheme not taking any postprocessing into account. The proof uses the results from Section 2, together with standard techniques for error estimates of the time discretization, so that the error bounds with respect to time are a direct consequence of the approximation properties of the Stokes projection operator. Using the post-processing approach previously discussed in the publication by Pan [loc. cit.] for the same spatial Stokes discretization, Theorem 3.2 finally provides a superconvergence result with respect to the energy norm for the postprocessed solution. In particular, second-order convergence in space for the postprocessed velocity gradient and the pressure norm is proven. The paper concludes with a short section on numerical experiments. The author uses the method of manufactured solution to demonstrate the expected rates of convergence in space and time by choosing \(\tau\sim h^2\). The analytical results, however, state that second-order convergence in time can only be observed for \(\tau\) being small enough and the author does not state clearly how \(\tau\) is chosen. Further, only studying this scaling does not demonstrate that both the spatial and the temporal rate of convergence are as expected. In particular, the error with respect to time discretization might be small enough to not contribute significantly even if it were of reduced order. At least with the chosen scaling time step size and mesh size, the results demonstrate the expected (super-)convergence results. Overall, the paper features a nice extension to the original results by Pan [loc. cit.] for the steady Stokes equations and is well written. Unfortunately, the restrictions to two dimensions and rectangular meshes are severe, making the approach not very useful in actual simulations. It would be very interesting to discuss if these results can be extended to at least the three-dimensional case and then also to the Navier-Stokes equations. Some more numerical examples with fixed time step sizes and mesh sizes would also have helped demonstrating the expected rates of convergence in the numerical examples better.
scientific article; zbMATH DE number 7468863
scientific article