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For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest positive eigenvalues, and the main challenge is that the coefficient matrix in the discrete Maxwell equation is indefinite and degenerate. To overcome this difficulty, we derive a singular value decomposition (SVD) of the discrete single-curl operator and then explicitly express the basis of the invariant subspace corresponding to the nonzero eigenvalues of the GEP. Consequently, we reduce the GEP to a null space free standard eigenvalue problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP and can be solved by the shift-and-invert Arnoldi method without being disturbed by the null space. Furthermore, the basis of the eigendecomposition is chosen carefully so that we can apply fast Fourier transformation (FFT)-based matrix vector multiplication to solve the embedded linear systems efficiently by an iterative method. For chiral and pseudochiral complex media, which are of great interest in magnetoelectric applications, the NFSEP can be further transformed to a null space free generalized eigenvalue problem whose coefficient matrices are Hermitian and Hermitian positive definite (HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method without shifting. Furthermore, the embedded linear system can be solved efficiently by using the conjugate gradient method without preconditioning and the FFT-based matrix vector multiplications. Numerical results are presented to demonstrate the efficiency of the proposed methods.

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