In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the L2-norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time dependent terms under the L2-norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and improve the superconvergence rates gradually. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be innity. Hence, we will seek some \hidden" condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical ndings.

In this paper, we propose an unfitted Nitsche's method to compute the band structures of phononic crystal with impurities of general geometry. The proposed method does not require the background mesh to fit the interfaces of impurities, and thus avoids the expensive cost of generating body-fitted meshes and simplifies the inclusion of interface conditions in the formulation. The quasi-periodic boundary conditions are handled by the Floquet-Bloch transform, which converts the computation of band structures into an eigenvalue problem with periodic boundary conditions. More importantly, we show the well-posedness of the proposed method using a delicate argument based on the trace inequality, and further prove the convergence by the Babuška-Osborn theory. We achieve the optimal convergence rate at the presence of the impurities of general geometry. We confirm the theoretical results by two numerical examples, and show the capability of the proposed methods for computing the band structures without fitting the interfaces of impurities.

Blind source separation (BSS) methods are useful tools to recover or enhance individual speech sources from their mixtures in a multi-talker environment. A class of efficient BSS methods are based on the mutual exclusion hypothesis of the source signal Fourier spectra on the timefrequency (TF) domain, and subsequent data clustering and classification. Though such methodology is simple, the discontinuous decisions in the TF domain for classification often cause defects in the recovered signals in the time domain. The defects are perceived as unpleasant ringing sounds, the so called musical noise. Post-processing is desired for further quality enhancement. In this paper, an efficient musical noise reduction method is presented based on a convex model of time-domain sparse filters. The sparse filters are intended to cancel out the interference due to major sparse peaks in the mixing coefficients or physically the early arrival and high energy portion of the room impulse responses. This strategy is efficiently carried out by l1 regularization and the split Bregman method. Evaluations by both synthetic and room recorded speech and music data show that our method outperforms existing musical noise reduction methods in terms of objective and subjective measures. Our method can be used as a post-processing tool for more general and recent versions of TF domain BSS methods as well.

In this work, we present an upscaled model for mixed dimensional coupled flow problem in fractured porous media. We consider both embedded and discrete fracture models (EFM and DFM) as fine scale models which contain coupled system of equations. For fine grid discretization, we use a conservative finite-volume approximation. We construct an upscaled model using the non-local multicontinuum (NLMC) method for the coupled system. The proposed upscaled model is based on a set of simplified multiscale basis functions for the auxiliary space and a constraint energy minimization principle for the construction of multiscale basis functions. Using the constructed NLMC-multiscale basis functions, we obtain an accurate coarse grid upscaled model. We present numerical results for both fine-grid models and upscaled coarse-grid models using our NLMC method. We consider model problems with (1) discrete fracture fine grid model with low and high permeable fractures;(2) embedded fine grid model for two types of geometries with differnet fracture networks and (3) embedded fracture fine grid model with heterogeneous permeability. The simulations using the upscaled model provide very accurate solutions with significant reduction in the dimension of the problem.

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.