Refined convergence structures of the rectangular Raviart-Thomas element

TL;DR

This paper investigates refined convergence properties of the rectangular Raviart-Thomas element for Laplace eigenvalue problems, including supercloseness, error expansions, and eigenvalue convergence from above.

Contribution

It introduces new supercloseness properties, error expansion formulas, and convergence proofs, enhancing understanding of the rectangular Raviart-Thomas element's performance.

Findings

Supercloseness enables improved post-processing accuracy.

Eigenvalues converge from above, verified by rigorous proofs.

Numerical experiments confirm theoretical results.

Abstract

In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. The essentially skillful method is the integral expansion for interpolation terms. Secondly, based on the supercloseness property, we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues, and provide a rigorous proof for them, based on which Richardson extrapolation can be performed. As a byproduct, we prove that all eigenvalues converge from above. Moreover, by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain. Thirdly, the equivalence between the lowest-order rectangular Raviart-Thomas element and the enriched rotated bilinear element is also indicated. At the last of this work, several numerical experiments are designed to demonstrate our theory.