The 2D Maxwell eigenproblems are studied from a new point of view. An electromagnetic problem is cast from the Lagrangian system with single variable into the Hamiltonian system with dual variables. The electric and magnetic components transverse to the wave propagation direction are treated as dual variables to each other. Higher order curl-conforming and divergence-conforming vector basis functions are used to construct dual vector spectral elements. Numerical examples demonstrate some unique advantages of the proposed method.

In this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ∈ provided only that ∈ ≤ N–1. An convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results.

A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

In this paper, nonconforming mixed finite element method is proposed to simulate the wave propagation in metamaterials. The error estimate of the semi-discrete scheme is given by convergence order O(h2), which is less than 40 percent of the computational costs comparing with the same effect by using Nédélec-Raviart element. A Crank-Nicolson full discrete scheme is also presented with O(τ2 + h2) by traditional discrete formula without using penalty method. Numerical examples of 2D TE, TM cases and a famous re-focusing phenomena are shown to verify our theories.

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number Re and magnetic Reynolds number Rm, or small δ=1–σ with

(C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that . Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

In this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy . An numerical example is also given to illustrate the effectiveness of the algorithm.

In this paper, we consider the transform magnetic (TM) model of electromagnetic scattering in the cavity. By the Polynomial Preserving Recovery technique, we present superconvergence analysis for the vertex-edge-face type finite element. From the numerical example, we can see that the provided method is efficient and stable.

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

This paper gives analysis of a semi-discrete scheme using equal order interpolation to solve unsteady Navier-Stokes equations. A unified pressure stabilized term is added to our scheme. We proved the uniform error estimates with respect to the Reynolds number, provided the exact solution is smooth.

In this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

A high-efficient algorithm to solve Crank-Nicolson scheme for variable coefficient parabolic problems is studied in this paper, which consists of the Function Time-Extrapolation Algorithm (FTEA) and Matrix Time-Extrapolation Algorithm (MTEA). First, FTEA takes a linear combination of previous l level solutions as good initial value of Un(see Time-extrapolation algorithm (TEA) for linear parabolic problems, J. Comput. Math., 32(2) (2014), pp. 183–194), so that Conjugate Gradient (CG)-iteration counts decrease to 1/3~1/4 of direct CG. Second, MTEA uses a linear combination of exact matrix values in level L, L+s, L+2s to predict matrix values in the following s–1 levels, and the coefficients of the linear combination is deduced by the quadric interpolation formula, then fully recalculate the matrix values at time level L+3s, and continue like this iteratively. Therefore, the number of computing the full matrix decreases by a factor 1/s. Last, the MTEA is analyzed in detail and the effectiveness of new method is verified by numerical experiments.

In this article, we consider the numerical solution for Poisson's equation in axisymmetric geometry. When the boundary condition and source term are axisymmetric, the problem reduces to solving Poisson's equation in cylindrical coordinates in the two-dimensional (r,z) region of the original three-dimensional domain S. Hence, the original boundary value problem is reduced to a two-dimensional one. To make use of the Mechanical quadrature method (MQM), it is necessary to calculate a particular solution, which can be subtracted off, so that MQM can be used to solve the resulting Laplace problem, which possesses high accuracy order and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by the splitting extrapolation algorithm (SEA). Meanwhile, a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.

This paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

In this work, the a-posteriori error indicator with an explicit formula for p-version finite element methods in square is investigated, and its reliable and efficient properties are deduced. Especially, this a-posteriori error indicator is determined by the right hand itemof themodel. We reformulate this a-posteriori error indicator with finite coefficients, which can be easily calculated during applications.

This paper is concerned with numerical solutions of the LDG method for 1D wave equations. Superconvergence and energy conserving properties have been studied. We first study the superconvergence phenomenon for linear problems when alternating fluxes are used. We prove that, under some proper initial discretization, the numerical trace of the LDG approximation at nodes, as well as the cell average, converge with an order 2k+1. In addition, we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points, respectively. As a byproduct, we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp. In the second part, we propose a fully discrete numerical scheme that conserves the discrete energy. Due to the energy conserving property, after long time integration, our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.