In this article, we consider the time-dependent Maxwell’s equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L ∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue.

A multilevel quality-guided phase unwrapping algorithm for real-time 3D shape measurement is presented. The quality map is generated from the gradient of the phase map. Multilevel thresholds are used to unwrap the phase level by level. Within the data points in each level, a fast scan-line algorithm is employed. The processing time of this algorithm is approximately 18.3 ms for an image size of 640480 pixels in an ordinary computer. We demonstrate that this algorithm can be implemented into our real-time 3D shape measurement system for real-time 3D reconstruction. Experiments show that this algorithm improves the previous scan-line phase unwrapping algorithm significantly although it reduces its processing speed slightly.

This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method based on a flexible type of non-polynomial local approximation space for the Helmholtz equation with varying wavenumber. The local approximation space consists of multiple polynomial-modulated phase functions which can be chosen according to the phase information of the solution. We obtain some {approximation} properties for this space and \textit{a prior} $L^2$ error estimates for the \textit{h}-convergence of the IPDG method using duality argument. We also provide ample numerical examples to show that, building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces.

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This paper is devoted to developing the Il'in-Allen-Southwell (IAS) parameter-uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection-diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct the IAS scheme for 1-D systems and then extend the scheme to 2-D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme is formally second order accurate in the sense that it is second order convergent when the perturbation parameter $\varepsilon$ is not too small and when the $\varepsilon$ is sufficiently small, the scheme is first order convergent in the discrete maximum norm uniformly in $\varepsilon$.