In this paper we develop and analyze the stochastic collocation method for solving the time-dependent Maxwell's equations with random coefficients and subject to random initial conditions. We provide a rigorous regularity analysis of the solution with respect to the random variables. To our best knowledge, this is the first theoretical results derived for the standard Maxwell's equations with random inputs. The rate of convergence is proved depending on the regularity of the solution. Numerical results are presented to confirm the theoretical analysis.

We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with $M$-matrix. It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm (SDA) with the shift-and-shrink transformation or the generalized Cayley transformation. In this paper, we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases. Meanwhile, the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined. Moreover, the convergence result and the comparison theorem on convergent rate are established. Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with $M$-matrix.

This paper is an attempt to investigate the nonlinear free vibration of skew plates reinforced by carbon nanotubes (CNTs) due to finite strain tensor. The material properties of the nano-composite are estimated using the molecular dynamic results and the rule of mixture. Also, the differential equations governing the motions are derived on the basis of Classical Plate Theory (CPT) regarding the nonlinear Green-Lagrange strain tensor. In order to solve the nonlinear equations, Galerkin's method, Frechet derivative and differential quadrature method are used. The effects of volume fraction of functionally graded materials (FGM), skew angle, distribution of CNTs and geometrical features of the plate on the nonlinear vibration of system have been studied. The results of this study have been compared with other researches and a good agreement has been achieved.

The $p$-Laplace problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):= ∫_ΩW(∇v)dx − ∫_Ωf vdx$ for $v∈W^{1,p}_0(Ω)$ with unique minimizer $u$ and stress $σ := DW(∇u)$. This paper proposes the discrete Raviart-Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix-Raviart nonconforming finite element method (CR-NCFEM). The sharper quasi-norm a priori and a posteriori error estimates of this two methods are presented. Numerical experiments are provided to verify the analysis.

In this paper the authors discuss the numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model is defined by an initial-boundary nonlinear convection-diffusion system of four partial differential equations: a parabolic equation for the pressure, two convection-diffusion equations for the concentrations of brine and radionuclide and a heat conduction equation for the temperature. The pressure appears within the concentration equations and heat conduction equation, and the Darcy velocity controls the concentrations and the temperature. The pressure is solved by the conservative mixed volume element method, and the order of the accuracy is improved by the Darcy velocity. The concentration of brine and temperature are computed by the upwind mixed volume element method on a changing mesh, where the diffusion is discretized by a mixed volume element and the convection is treated by an upwind scheme. The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy. The mixed volume element has the local conservation of mass and energy, and it can obtain the brine and temperature and their adjoint vector functions simultaneously. The conservation nature plays an important role in numerical simulation of underground fluid. The concentrations of radionuclide factors are solved by the method of upwind fractional step difference and the computational work is decreased by decomposing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup. By the theory and technique of a priori estimates of differential equations, we derive an optimal order result in $L^2$ norm. Numerical examples are given to show the effectiveness and practicability and the composite method is testified as a powerful tool to solve the well-known actual problem.

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory (WENO) finite difference schemes. Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems, and the two-dimensional Rayleigh-Taylor instability (RTI) problem as examples, we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition, which are often given and used in the literature, and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem. An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error. The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability, reduced rounding errors and increased computational efficiency.

In this paper, we study a class of nonlinear fractional integro-differential equations. The fractional derivative is described in the Caputo sense. Using the properties of the Caputo derivative, we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel, then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra type. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in $L^∞$-norm and weighted $L^2$-norm.

The application of the singular boundary method (SBM), a relatively new meshless boundary collocation method, to the inverse Cauchy problem in three-dimensional (3D) linear elasticity is investigated. The SBM involves a coupling between the non-singular boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS. Due to the boundary-only discretizations and its semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique, while the optimal regularization parameter is determined by the $L$-curve criterion. Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.

The electrokinetic mixing, as a powerful technique in microfluidic devices, is widely used in many applications. In this study, a more general dynamic model, which consists of Poisson equation, Nernst-Planck equation and Navier-Stokes equations, is used to describe the electrokinetic mixing of non-Newtonian fluids in microchannels. Furthermore, a coupled multiple-relaxation-time (MRT) lattice Boltzmann (LB) framework is developed to solve this complicated multi-physics transport phenomenon. In numerical simulations, we mainly consider the effects of the arrangement of nonuniform surface potentials and the power-law index on the mixing efficiency and the volumetric flow rate. Numerical results show that the mixing efficiency and the volumetric flow rate of shear-thinning fluids are higher than that of shear-thickening fluids under the same condition. It is also shown that for both types of fluids, there should be a balance between the mixing and liquid transport in electrokinetic microfluidics.

The cold fluid flowing over two hot spheroids placed in a tandem arrangement was numerically studied via a GPU-based immersed boundary-lattice Boltzmann method (IB-LBM) model. The drag coefficient and average Nusselt number of both the two spheroids were obtained with the main influencing factors investigated. To validate the IB-LBM model, several numerical case studies containing one and two spheres were firstly conducted to reach the good agreement with the previously reported data. Then, a number of simulations were further carried out which were designed by changing the particle aspect ratio (1.0≤ $Ar$ ≤4.0) and inter particle distance (1.5≤ $l$ ≤7.0, where $l = L/D$, $D$ stands for the volume-equivalent sphere diameter) as well as the Reynolds number (10 ≤ $Re$ ≤200). Their influence on the momentum and heat transfer characteristics between the solid and fluid phases were fully discussed. Numerical results show that, for all the considered Reynolds numbers and aspect ratios, the individual and total drag coefficients and average Nusselt number increase with the inter particle distance. The inter particle distance has greater influence on the drag coefficient and average Nusselt number of the trailing particle than the leading one. The drag coefficient and average Nusselt number of the trailing particle are far less than the leading one under the same working conditions. The prediction correlations for the drag coefficient and average Nusselt number of both the two spheroids were established with low deviations. At last, the influence of the relative incidence angles between the two tandem spheroids on the momentum and heat transfer was studied. It is shown that the relative incidence angles play significant roles due to the change of the frontal area of the leading spheroid with these angles.

In this paper, we combine the generalized multiscale finite element method (GMsFEM) with the balanced truncation (BT) method to address a parameter-dependent elliptic problem. Basically, in progress of a model reduction we try to obtain accurate solutions with less computational resources. It is realized via a spectral decomposition from the dominant eigenvalues, that is used for an enrichment of multiscale basis functions in the GMsFEM. The multiscale bases computations are localized to specified coarse neighborhoods, and follow an offline-online process in which eigenvalue problems are used to capture the underlying system behaviors. In the BT on reduced scales, we present a local-global strategy where it requires the observability and controllability of solutions to a set of Lyapunov equations. As the Lyapunov equations need expensive computations, the efficiency of our combined approach is shown to be readily flexible with respect to the online space and an reduced dimension. Numerical experiments are provided to validate the robustness of our approach for the parameter-dependent elliptic model.