The paper is devoted to solving the boundary value problems with random parameters. We consider flows in a porous medium with random permeability field or random boundary conditions. The Monte Carlo method with double randomization is suggested to compute the statistical properties of the flow. The paper focuses on the calculating of media’s effective permeability. The method is compared with a standard Monte Carlo approach. Numerical tests show that double randomization gives high accuracy and can improve computational efficiency.

The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as well as quantitative point of view. We construct useful bounds of solution yielding estimates for the optimal value of the stock to bond proportion in the portfolio. Furthermore, we construct asymptotic expansions of a solution in terms of a small model parameter. Finally, we perform sensitivity analysis of the optimal solution with respect to various model parameters and compare analytical results of this paper with the corresponding known results arising from time-discrete dynamic stochastic optimization model.

In this paper, we establish the convergence of a nonconforming triangular Morley element for the plate bending problem on degenerate meshes. An explicit bound for the interpolation error is derived for arbitrary triangular meshes without any assumptions. The optimal convergence rates of the moment error and rotation error are derived for triangular meshes satisfying the maximal angle condition. Our results can also be extended to the three dimensional Morley element presented recently in [41]. Finally, some numerical results are reported that confirm our theoretical results.

Local projection based stabilized finite element methods for the solution of Darcy flow offer several advantages as compared to mixed Galerkin methods. In particular, the avoidance of stability conditions between finite element spaces, the efficiency in solving the reduced linear algebraic system, and the convenience of using equal order continuous approximations for all variables. In this paper we analyze the pressure gradient method for Darcy flow and investigate its stability and convergence properties.

We discuss an optimal control problem of laser surface hardening of steel which is governed by a dynamical system consisting of a semilinear parabolic equation and an ordinary differential equation with a non differentiable right hand side function $f_+$. To avoid the numerical and analytic difficulties posed by $f_+$, it is regularized using a monotone Heaviside function and the regularized problem has been studied in literature. In this article, we establish the convergence of solution of the regularized problem to that of the original problem. The estimates, in terms of the regularized parameter, justify the existence of solution of the original problem. Finally, a numerical experiment is presented to illustrate the effect of regularization parameter on the state and control errors.

In this paper we analyze the Local Discontinuous Galerkin (LDG) method for the constrained optimal control problem governed by the unsteady convection diffusion equations. A priori error estimates are obtained for both the state, the adjoint state and the control. For the discretization of the control we discuss two different approaches which have been used for elliptic optimal control problem.

In this paper, we first extend a simple algorithm proposed by Jia et al. [16] to color/vectorial images, and then apply the vectorial algorithm to some variational models for image restoration problems including color image denoisings with the red-green-blue (RGB) and chromaticity-brightness (CB) color representations, CB based colorization and image inpainting. The variational models are all total variation (TV)-based. The proposed vectorial algorithm is simple and straightforward to implement. Some numerical experiments show that it is fast and efficient.

We consider a two-point boundary-value problem for a singularly perturbed convection-diffusion problem. The problem is solved by using a defect-correction method based on a first-order upwind difference scheme and a second-order (unstabilized) central difference scheme.
A robust a posteriori error estimate in the maximum norm is derived. It provides computable and guaranteed upper bounds for the discretization error. Numerical examples are given that illustrate the theoretical findings and verify the efficiency of the error estimator on a priori adapted meshes and in an adaptive mesh movement algorithm.

In this paper, we propose a subgrid finite element method for the two-dimensional (2D) stationary incompressible Navier-Stokes equation (NSE) based on high order finite element polynomial interpolations. This method yields a subgrid eddy viscosity which does not act on the large scale flow structures. The proposed eddy viscous term consists of the fluid flow fluctuation stress. The fluctuation stress can be calculated by means of simple reduced-order polynomial projections. Assuming some regular results of NSE, we give a complete error analysis. Finally, in the part of numerical tests, the numerical computations show that the numerical results agree with some benchmark solutions and theoretical analysis very well.

This paper deals with an interaction problem between a solid and an electromagnetic field in the frequency domain. More precisely, we aim to compute both the magnetic component of the scattered wave and the elastic vibrations that take place in the solid elastic body. To this end, we solve a transmission problem holding between the bounded domain $\Omega_S \subset R^3$ representing the obstacle and a sufficiently large annular region surrounding it. We point out here that (following Voigt's model, cf. [12]) we only allow the electromagnetic field to interact with the elastic body through the boundary of $\Omega_S$. We apply the abstract framework developed in the work [3] by A. Buffa to prove that our coupled variational formulation is well posed. We define the corresponding discrete scheme by using the edge element in the electromagnetic domain and standard Lagrange finite elements in the solid domain. Then we show that the resulting Galerkin scheme is uniquely solvable, convergent and we derive optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments.

A new nonconforming element constructed by the Double Set Parameter method, is applied to the fourth order elliptic singular perturbation problem. The convergence uniformly in the perturbation parameter $\varepsilon$, is proved under the anisotropic meshes and optimal convergence rate $O(h)$ is obtained. Numerical results are given to demonstrate validity of our theoretical analysis.

In this paper, the $\theta$ scheme of operator splitting methods is applied to the Navier-Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind with the Navier-Stokes operator. Firstly, we introduce the multiplier such that the variational inequality is equivalent to the variational identity. Subsequently, we give the $\theta$ scheme to compute the variational identity and consider the finite element approximation of the $\theta$ scheme. The stability and convergence of the $\theta$ scheme are showed. Finally, we give the numerical results.