We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

This paper is devoted to the study of the Eulerian-Lagrangian method (ELM) for convection-diffusion equations on unstructured grids with or without accurate numerical integration. We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method. Our approach is based on an algorithm for finding the intersection of two non-matching grids. It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible. The evaluation of the integrals leads to increased precision and the unconditional stability. We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features: firstly, it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly, the overall cost of the proposed method is comparable with the traditional ones.

In this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer (PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of $H(grad)$ variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete $\boldsymbol{H}(curl)$-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use $H$-$X$ preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.

It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes, such as nano-hydrodynamics, drop wise condensation, oceanography, earthquake and wind ripple in desert. Inspired and motivated by these facts, we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations. The proposed technique is applied without any discretization, perturbation, transformation, restrictive assumptions and is free from Adomian's polynomials. Several examples are given to verify the reliability and efficiency of the proposed technique.

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.

The pullback asymptotic behavior of the solutions for 2D Nonautonomous G-Navier-Stokes equations is studied, and the existence of its $L^2$-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.

In this paper, we propose a new two-level preconditioned C-G method which uses the quadratic smoothing and the linear correction in distorted but topologically structured grid. The CPU time of this method is less than that of the multigrid preconditioned C-G method (MGCG) using the quadratic element, but their accuracy is almost the same. Numerical experiments and eigenvalue analysis are given and the results show that the proposed two-level preconditioned method is efficient.

In this paper, exact vibration frequencies of circular, annular and sector membranes with a radial power law density are presented for the first time. It is found that in general, the sequence of modes may not correspond to increasing azimuthal mode number $n$. The normalized frequency increases with the absolute value of the power index $|ν|$. For a circular membrane, the fundamental frequency occurs at $n = 0$ where $n$ is the number of nodal diameters. For an annular membrane, the frequency increases with respect to the inner radius $b$. When $b$ is close to one, the width $1 − b$ is the dominant factor and the differences in frequencies are small. For a sector membrane, $n − 1$ is the number of internal radial nodes and the fundamental frequency occurs at $n = 1$. Increased opening angle $β$ increases the frequency.