In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation.

Solutions for many problems of interest exhibit singular behaviors at domain corners or points where boundary condition changes type. For this type of problems, direct spectral methods with usual polynomial basis functions do not lead to a satisfactory convergence rate. We develop in this paper a M\"untz-Galerkin method which is based on specially tuned M\"untz polynomials to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and M\"untz polynomials, we develop efficient implementation procedures for the M\"untz-Galerkin method, and provide optimal error estimates. As examples of applications, we consider the Poisson equation with mixed Dirichlet-Neumann boundary conditions, whose solution behaves like $O(r^{1/2})$ near the singular point, and demonstrate that the M\"untz-Galerkin method greatly improves the rates of convergence of the usual spectral method.

In this paper, we propose a novel iterative convolution-thresholding method (ICTM) that is applicable to a range of variational models for image segmentation. A variational model usually minimizes an energy functional consisting of a fidelity term and a regularization term. In the ICTM, the interface between two different segment domains is implicitly represented by their characteristic functions. The fidelity term is usually written as a linear functional of the characteristic functions and the regularized term is approximated by a functional of characteristic functions in terms of heat kernel convolution. This allows us to design an iterative convolution-thresholding method to minimize the approximate energy. The method is simple, efficient and enjoys the energy-decaying property. Numerical experiments show that the method is easy to implement, robust and applicable to various image segmentation models.

This work is devoted to the numerical computation of complex band structure $\k=\k(\omega)\in\mathbb C^3$ for positive $\omega$ of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in $\k\in\mathbb C^3$ and to view the remaining one as the eigenvalue of a quadratic operator pencil derived from Maxwell's equations. Then Yee's scheme is employed to discretize $\nabla\times$ and $\k\times$ operators in this quadratic operator pencil. Distinct from the others' works which either ignore or directly exploit the Hamiltonian structure of the spectrum of the resulting QEP, we reformulate this QEP into an equivalent $\top$-palindromic QEP to facilitate the use of superior structure-preserving algorithms. Ultimately we rely on the structured Arnoldi algorithm, namely the G$\top$SHIRA algorithm, to compute eigenvalues of a $\top$-skew-Hamiltonian pair which are near or in $[-2,2]$, a much narrower region than the whole positive real axis in the origin problem. Moreover, to accelerate the inner iterations of the G$\top$SHIRA algorithm, we propose the preconditioning technique, making most of the eigenmatrix, which can essentially be seen as the Kronecker product of three discrete Fourier transformation matrices, of the commutative discretized $\partial_x,\partial_y,\partial_z$ operators. The advantage of our method is discussed in detail and corroborated by several numerical results.

Local discontinuous Galerkin (LDG) methods are popular for convection-diffusion equations. In LDG methods, we introduce an auxiliary variable $p$ to represent the derivative of the primary variable $u$, and solve them on the same mesh. In this paper, we will introduce a new LDG method, and solve $u$ and $p$ on different meshes. The stability and error estimates will be investigated. The new algorithm is more flexible and flux-free for pure diffusion equations without introducing additional computational cost compared with the original LDG methods, since it is not necessary to solve each equation twice. Moreover, it is possible to construct third-order maximum-principle-preserving schemes based on the new algorithm. However, one cannot anticipate optimal accuracy in some special cases. In this paper, we will find out the reason for accuracy degeneration which further leads to several alternatives to obtain optimal convergence rates. Finally, several numerical experiments will be given to demonstrate the stability and optimal accuracy of the new algorithm.