We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

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.

This article summaries a numerical study of thermo-solutal natural convection in a square cavity filled with anisotropic porous medium. The side walls of the cavity are maintained at constant temperatures and concentrations, whereas bottom wall is a function of non-uniform (sinusoidal) temperature and concentration. The non-Darcy Brinkmann model is considered. The governing equations are solved numerically by spectral element method using the vorticity-stream-function approach. The controlling parameters for present study are Darcy number (Da), heat source intensity i.e., thermal Rayleigh number (Ra), permeability ratio (K*), orientation angle (ϕ). The main attention is given to understand the impact of anisotropy parameters on average rates of heat transfer (bottom, Nub, side Nus) and mass transfer (bottom, Shb, side, Shs) as well as on streamlines, isotherms and iso-concentration. Numerical results show that, for irrespective value of K*, the heat and mass transfer rates are negligible for 10−7 ≤ Da ≤ 10−5, Ra = 2 × 105 and ϕ = 45°. However a significant impact appears on Nusselt and Sherwood numbers when Da lies between 10−5 to 10−4. The maximum bottom heat and mass transfer rates (Nub, Sub) is attained at ϕ = 45°, when K* =0.5 and 2.0. Furthermore, both heat and mass transfer rates increase on increasing Rayleigh number (Ra) for all the values of K*. Overall, It is concluded from the above study that due to anisotropic permeability the flow dynamics becomes complex.

It is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

This work pertains to numerical aspects of a finite element method based discontinuous functions. Our study focuses on the Interior Penalty Discontinuous Galerkin method (IPDGM) because of its high-level of flexibility for solving the full wave equation in heterogeneous media. We assess the performance of IPDGM through a comparison study with a spectral element method (SEM). We show that IPDGM is as accurate as SEM. In addition, we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

In this paper, we study the numerical solution of the Stokes system in deformed axisymmetric geometries. In the azimuthal direction the discretization is carried out by using truncated Fourier series, thus reducing the dimension of the problem. The resulting two-dimensional problems are discretized using the spectral element method which is based on the variational formulation in primitive variables. The meridian domain is subdivided into elements, in each of which the solution is approximated by truncated polynomial series. The results of numerical experiments for several geometries are presented.

Spectral element methods are well established in the field of wave propagation, in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.