This is an extension of our earlier work [9] in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. An algorithm of building a suitable polygonal mesh based on the random points was given and the traditional discontinuous Galerkin (DG) method was adopted on the constructed polygonal mesh. Numerical results in [9] show that the current scheme will generate spurious numerical oscillations when dealing with solutions containing strong shocks. In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for DG schemes on two-dimensional unstructured triangular meshes [27], to our high order method on polygonal meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the original method in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt is particularly simple to implement and will not harm the conservativeness and compactness of the original method. Moreover, we also extend the maximum-principle-satisfying limiter for the scalar case and the positivity-preserving limiter for the Euler system to our method. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of these limiters.

We investigate an $h$-$p$ version of the continuous Petrov-Galerkin method for the nonlinear Volterra functional integro-differential equations with vanishing delays. We derive $h$-$p$ version a priori error estimates in the $L^2$-, $H^1$- and $L^∞$-norms, which are completely explicit in the local discretization and regularity parameters. Numerical computations supporting the theoretical results are also presented.

In this paper, we consider a multi-layer porous wall model for coronary drug-eluting stents that leads to an interface problem whose coefficients have multiple discontinuous points, and an imperfect contact interface jump condition is imposed at the first discontinuous point where the stent meets the artery. The existence and uniqueness of the solution to the related weak problem are established. A linear and a quadratic immersed finite element (IFE) methods are developed for solving this interface problem. Error estimation is carried out to show that the proposed IFE methods converge optimally. Numerical examples are presented to demonstrate features of these IFE methods.

The dust aerosols have an important effect on the solar radiation in the Martial atmosphere and both surface and atmospheric heating rates, which are also basic drivers of atmospheric dynamics. Aerosols cause an attenuation of the solar radiation traversing the atmosphere and this attenuation is modeled by the Lambert-Beer-Bouguer law, where the aerosol optical thickness plays an important role. Through Angstrom law, the aerosol optical thickness can be approximated and this law allows to model attenuation of the solar radiation traversing the atmosphere by a fractional diffusion equation. The analytical solution is available in the case of one space dimension. When we extend the fractional diffusion equation to the case of two or more space variables, we need large and massive computations to approach numerically the solutions. In this case a suitable strategy is to use the cloud computing to carry out the simulations. We present an introduction to cloud computing applied to the fractional diffusion equation in one dimension.

A nonoverlapping domain decomposition method (DDM) is proposed to solve general elastic body-plate problems, discretized by the $P_1$-NZT finite element method. It is proved in a subtle way that the convergence rate of the method is optimal (independent of the finite element mesh size), even for a regular family of finite element triangulations. This enables us to combine the method with adaptive techniques in practical applications. Some numerical results are included to illustrate the computational performance of the method.

The singularity-separated method (SSM) for the singular perturbation problem $-\epsilon u''+bu' + cu = f(x), u(0) = u(1) = 0$, is proposed for the first time. The solution is expressed as $u = w-ν$, where $w$ is the solution of corresponding third boundary value problem and $ν$ is an exact singular function. We have proved a global regularity, $||w||_2 ≤ C$, where the constant $C$ is independent of $\epsilon$, and discussed three kinds of finite element (FE) methods with SSM. Numerical results show that these FE-solutions have the high accuracy when only one element in boundary layer is taken.

In this paper, we implement and analyse a spectral element method (SEM) on hybrid triangular and quadrilateral element meshes, where the elemental transformation between the triangular element and the reference element is based on the mapping in [17]. We introduce the notion of "quasi-interpolation" to glue the hybrid elements which can build in the singularity of the elemental mapping, and only affects one coefficient of the tensorial nodal basis expansion. Therefore, the hybrid method can be implemented as efficiently as the usual quadrilateral SEM. We also rigorously analyse the "quasi-interpolation" error and the convergence of the hybrid SEM, which show the spectral accuracy can be kept.

In this paper, an asymptotic preserving implicit unified gas kinetic scheme (IUGKS) is constructed for the frequency-dependent radiative transfer equations. Different from the asymptotic preserving unified gas kinetic scheme (UGKS) which uses the explicit initial value of the radiation intensity in the construction of the boundary fluxes as in the previous works [Sun et al., J. Comput. Phys. 285 (2015), pp. 265-279 and J. Comput. Phys. 302 (2015), pp. 222-238], here we construct the boundary fluxes by a back-time discretization so that they depend implicitly on the radiation intensity. Thus, the time step constraint by the Courant-Friedrichs-Lewy (CFL) condition is not needed anymore for IUGKS. It is shown that IUGKS is asymptotic preserving uniformly with the small Knudsen parameter. A number of numerical tests have been carried out and the numerical results show that large time steps can be used for the current scheme, and the computational efficiency can be improved greatly in comparison with UGKS and the implicit Monte Carlo scheme.

We introduce an optimization model of the support vector regression with the group lasso regularization and develop a class of efficient two-step fixed-point proximity algorithms to solve it numerically. To overcome the difficulty brought by the non-differentiability of the group lasso regularization term and the loss function in the proposed model, we characterize its solutions as fixed-points of a nonlinear map defined in terms of the proximity operators of the functions appearing in the objective function of the model. We then propose a class of two-step fixed-point algorithms to solve numerically the optimization problem based on the fixed-point equation. We establish convergence results of the proposed algorithms. Numerical experiments with both synthetic data and real-world benchmark data are presented to demonstrate the advantages of the proposed model and algorithms.

In this paper, we study the well-posedness of a fractional Stokes equation and its numerical solution. We first establish the well-posedness of the weak problem by suitably define the fractional Laplacian operator and associated functional spaces. The existence and uniqueness of the weak solution is proved by using the classical saddle-point theory. Then, based on the proposed variational framework, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: 1) a theoretical framework for the variational solutions of the fractional Stokes equation; 2) an efficient spectral method for solving the weak problem, together with a detailed numerical analysis providing useful error estimates for the approximative solution; 3) a fast implementation technique for the proposed method and investigation of the discrete system. Finally, some numerical experiments are carried out to confirm the theoretical results.

In this survey article, we present some recent results on the asymptotic behavior of four systems of orthogonal polynomials. These are Stieltjes-Wigert, Hahn, Racah and pseudo-Jacobi polynomials. In each case, the variable $z$ is allowed to be in any part of the complex plane. In some cases, asymptotic formulas are also given for their zeros.

We study uniform bounds associated with the Allen-Cahn equation and its numerical discretization schemes. These uniform bounds are different from, and weaker than, the conventional energy dissipation and the maximum principle, but they can be helpful in the analysis of numerical methods. In particular, we show that finite difference spatial discretization, like the original continuum model, shares the uniform $L^p$-bound for all even $p$, which also leads to the maximum principle. In comparison, a couple of other spatial discretization schemes, namely the Fourier spectral Galerkin method and spectral collocation method preserve the $L^p$-bound only for $p = 2$. Moreover, fully discretized schemes based on the Fourier collocation method for spatial discretization and Strang splitting method for time discretization also preserve the uniform $L^2$-bound unconditionally.

We propose a family of conforming mixed triangular prism finite elements for solving the classical Hellinger-Reissner mixed problem of the linear elasticity equations in three dimensions. These elements are constructed by product of elements on triangular meshes and elements in one dimension. The well-posedness is established for all elements with $k ≥ 1$, which are of $k+1$ order convergence for both the stress and displacement. Besides, a family of reduced stress spaces is proposed by dropping the degrees of polynomial functions associated with faces. As a result, the lowest order conforming mixed triangular prism element has 93 plus 33 degrees of freedom on each element.

Fully diagonalized Chebyshev spectral methods for solving second and fourth order elliptic boundary value problems are proposed. They are based on appropriate base functions for the Galerkin formulations which are complete and biorthogonal with respect to certain Sobolev inner product. The suggested base functions lead to diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy.

This paper is concerned with the computable error estimates for the eigenvalue problem which is solved by the general conforming finite element methods on the general meshes. Based on the computable error estimate, we can give an asymptotically lower bound of the general eigenvalues. Furthermore, we also give a guaranteed upper bound of the error estimates for the first eigenfunction approximation and a guaranteed lower bound of the first eigenvalue based on computable error estimator. Some numerical examples are presented to validate the theoretical results deduced in this paper.

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and ill-conditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver for this preconditioner in multi-dimension. A modified preconditioner based on the approximate inverse technique is constructed in this paper. The computational cost of each iteration in solving the preconditioned system is $\mathcal{O}(\ell N_x N_y log N_x)$, where $N_x$, $N_y$ are the grid sizes in each direction and $\ell$ is a small integer. Numerical examples are given to demonstrate the efficiency of the proposed preconditioner.

We establish in this paper the supercloseness of the quadratic finite element solution of a two dimensional elliptic problem to the piecewise quadratic interpolation of its exact solution. The assumption is that the partition of the solution domain is quasi-uniform under a Riemannian metric and that each pair of the adjacent elements in the partition forms an approximate parallelogram. This result extends our previous one in [7] for the linear finite element approximations based on adaptively refined anisotropic meshes. It also generalizes the results by Huang and Xu in [13] for the supercloseness of the quadratic elements based on the mildly structured quasi-uniform meshes. A distinct feature of our analysis is that we transform the error estimates on each physical element to that on an equilateral standard element, and then focus on the algebraic properties of the Jacobians of the affine mappings from the standard element to the physical elements. We believe this idea is also useful for the superconvergence study of other types of elements on unstructured meshes.