This paper presents a GPU-accelerated implementation of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method with an inexact nullspace filtering approach to find eigenvalues in electromagnetics analysis with higher-order FEM. The performance of the proposed approach is verified using the Kepler (Tesla K40c) graphics accelerator, and is compared to the performance of the implementation based on functions from the Intel MKL on the Intel Xeon (E5-2680 v3, 12 threads) central processing unit (CPU) executed in parallel mode. Compared to the CPU reference implementation based on the Intel MKL functions, the proposed GPU-based LOBPCG method with inexact nullspace filtering allowed us to achieve up to 2.9-fold acceleration.

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 propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with imperfect contact in two dimensions, which has not been well-studied in the literature. Numerical experiments demonstrated the effectiveness of our method.

This study makes the first attempt to accelerate the singular boundary method (SBM) by the precorrected-FFT (PFFT) for large-scale three-dimensional potential problems. The SBM with the GMRES solver requires computational complexity, where N is the number of the unknowns. To speed up the SBM, the PFFT is employed to accelerate the SBM matrix-vector multiplication at each iteration step of the GMRES. Consequently, the computational complexity can be reduced to . Several numerical examples are presented to validate the developed PFFT accelerated SBM (PFFT-SBM) scheme, and the results are compared with those of the SBM without the PFFT and the analytical solutions. It is clearly found that the present PFFT-SBM is very efficient and suitable for 3D large-scale potential problems.

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Hexagonal grids are valuable in two-dimensional applications involving Laplacian. The methods and analysis are investigated in current work in both linear and nonlinear problems related to anisotropic Laplacian. Ordinary and compact hexagonal grid finite difference methods are developed by elementary arguments, and then analyzed by perturbation for standard Laplacian. In the anisotropic case, analysis is done through reduction to the standard one by using Fourier vectors of mixed types. These hexagonal seven-point methods, with established theoretic stabilities and accuracies, are numerically confirmed in linear and semi-linear anisotropic Poisson problems, and can be applied also in time-dependent problems and in many applications in two-dimensional irregular domains.

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, form a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

A coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

We study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave number k are derived. This shows that the pollution error in the broken H1-norm is bounded by under mesh condition k7/2h2≤C0 or (kh)2+k(kh)p+1≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Here h is the mesh size, p is the order of the approximation space and C0 is a constant independent of k and h. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

In this paper, we study the role of mesh quality on the accuracy of linear finite element approximation. We derive a more detailed error estimate, which shows explicitly how the shape and size of elements, and symmetry structure of mesh effect on the error of numerical approximation. Two computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the mesh. In compare with the standard a priori error estimates, which only yield information on the asymptotic error behaviour in a global sense, our proposed error estimate considers the effect of local element geometry properties, and is thus more accurate. Under certain conditions, the traditional error estimates and supercovergence results can be derived from the proposed error estimate. Moreover, the estimators Ge and Gv are computable and thus can be used for predicting the variation of errors. Numerical tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

We investigate the defect structures around a spherical colloidal particle in a cholesteric liquid crystal using spectral method, which is specially devised to cope with the inhomogeneity of the cholesteric at infinity. We pay particular attention to the cholesteric counterparts of nematic metastable configurations. When the spherical colloidal particle imposes strong homeotropic anchoring on its surface, besides the well-known twisted Saturn ring, we find another metastable defect configuration, which corresponds to the dipole in a nematic, without outside confinement. This configuration is energetically preferable to the twisted Saturn ring when the particle size is large compared to the nematic coherence length and small compared to the cholesteric pitch. When the colloidal particle imposes strong planar anchoring, we find the cholesteric twist can result in a split of the defect core on the particle surface similar to that found in a nematic liquid crystal by lowering temperature or increasing particle size.

A class of finite volume methods is developed for pricing either European or American options under jump-diffusion models based on a linear finite element space. An easy to implement linear interpolation technique is derived to evaluate the integral term involved, and numerical analyses show that the full discrete system matrices are M-matrices. For European option pricing, the resulting dense linear systems are solved by the generalised minimal residual (GMRES) method; while for American options the resulting linear complementarity problems (LCP) are solved using the modulus-based successive overrelaxation (MSOR) method, where the H+-matrix property of the system matrix guarantees convergence. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of these methods.

Admissible regions for higher-order finite volume method (FVM) grids are considered. A new Hermite quintic FVM and a new hybrid quintic FVM are constructed to solve elliptic boundary value problems, and the corresponding admissible regions are investigated. A sufficient condition for the uniform local-ellipticity of the new hybrid quintic FVM is obtained when its admissible region is known. In addition, the admissible regions for a large number of higher-order FVMs are provided. For the same class of FVM (Lagrange, Hermite or hybrid), the higher order FVM has a smaller admissible region such that stronger geometric restrictions are required to guarantee its uniform local-ellipticity.

We construct modulus-based synchronous multisplitting iteration methods to solve a large implicit complementarity problem on parallel multiprocessor systems, and prove their convergence. Numerical results confirm our theoretical analysis and show that these new methods are efficient.

An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.

Pressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.

We present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

A cheapest stable nonconforming finite element method is presented for solving the incompressible flow in a square cavity without smoothing the corner singularities. The stable cheapest nonconforming finite element pair based on P1×P0 on rectangularmeshes [29] is employed with a minimal modification of the discontinuous Dirichlet data on the top boundary, where is the finite element space of piecewise constant pressures with the globally one-dimensional checker-board pattern subspace eliminated. The proposed Stokes elements have the least number of degrees of freedom compared to those of known stable Stokes elements. Three accuracy indications for our elements are analyzed and numerically verified. Also, various numerous computational results obtained by using our proposed element show excellent accuracy.