A posteriori and superconvergence error analysis for finite element approximation of the Steklov eigenvalue problem

In the current paper, we introduce an error analysis method and a new procedure to accelerate the convergence of finite element (FE) approximation of the Steklov eigenvalue problem. The error analysis consists of three steps. First, we introduce an optimal residual type the a posteriori error estimator, and prove its efficiency and reliability. Next, we present a residual type the a priori estimate in terms of derivatives of the eigenfunctions. Finally, we prove accurate the a priori error estimates by combining the a priori residual estimate and the a posteriori error estimates. The new procedure for accelerating the convergence comes from a postprocessing technique, in which we solve an auxiliary source problem on argument spaces. The argument space can be obtained similarly as in the two-space method by increasing the order of polynomials by one. We end the paper by reporting the results of a couple of numerical tests, which allow us to assess the performance of the new error analysis and the postprocessing method.