We start with a three-dimensional equilibrium problem involving a linearly elastic solid at small strains subjected to unilateral contact conditions. The reference configuration of the solid is assumed to be a thin shallow shell with a uniform thickness. We focus on the limit when the thickness tends to zero, i.e. when the three-dimensional domain tends to a two-dimensional one. In the generic case, this means that the initial Signorini problem, where the contact conditons hold on the boundary, tends to an obstacle problem, where the contact conditions hold in the domain. When the problem is stated in terms of curvilinear coordinates, the unilateral contact conditions involve a non penetrability inequality which couples the three covariant components of the displacement. We show that nevertheless we can uncouple these components and the contact conditions involve only the transverse covariant component of the displacement at the limit.

In this paper, we consider a class of new accelerated multilevel aggregation methods using two polynomial-type vector extrapolation methods, namely the reduced rank extrapolation (RRE) and the generalization of quadratic extrapolation (GQE) methods. We show how to combine the multilevel aggregation methods with the RRE and GQE algorithms on the finest level in order to speed up the numerical computation of the stationary probability vector for an irreducible Markov chain. Numerical experiments on typical Markov chain problems are reported to illustrate the efficiency of the accelerated multilevel aggregation methods .

Two-level finite element methods are applied to solve numerically the 2D ⁄ 3D steady Navier-Stokes equations if a strong uniqueness condition (\frac{\|f\|_{-1}}{\|f\|_0})^\frac{1}{2}\leq\delta=1-\frac{N\|F\|_{-1}}{\nu^2} holds, where N is defined in (2.4)-(2.6). Moreover, one-level finite element method is applied to solve numerically the 2D/3D steady Navier-Stokes equations if a weak uniqueness condition 0<\delta<(\frac{\|f\|_{-1}}{\|f\|_0})^\frac{1}{2} holds. The two-level algorithms are motivated by solving a nonlinear problem on a coarse grid with mesh size H and computing the Stokes, Oseen and Newton correction on a fine grid with mesh size h << H. The uniform stability and convergence of these methods with respect to  and grid sizes h and H are provided. Finally, some numerical tests are made to demonstrate the effectiveness of one-level method and the three two-level methods.

This study considers Pao's transfer theory of turbulence for the family of Approximate Deconvolution Models (ADMs). By taking a different representation of the persistent input of energy into the large scales of the turbulent flow, the Pao theory simplifies somewhat. Analysis of the resulting model is given and it is verified that (after the simplification as was known before it) it is consistent with the important statistics of homogeneous isotropic turbulence. The ADMs have an enhanced energy dissipation and a modification to the kinetic energy which affect the truncation of scales by reducing the models microscale from the Kolmogorov microscale. The energy dissipation can be even more enhanced by the time relaxation and the effects of this term are presented as well.

The high dimensional model representation (HDMR) method was recently proposed as an efficient tool to capture the input-output relationships in high-dimensional systems for many problems in science and engineering. In this paper, we develop a new multiple sub-domain random sampling HDMR method (MSD-RS-HDMR) for general high dimensional input-output systems. The domain splitting technique is applied to divide the whole domain into multiple sub-domains. The RS-HDMR method is used to generate local approximations in sub-domains and a set of weight functions is introduced to obtain approximations near the sub-domain interfaces. Numerical experiments are carried out using the MSD-RS-HDMR method for given high dimensional functions and a real application of the Tropospheric Alkane Photochemistry model. The new method has been demonstrated to be very effective, and numerical results confirm the excellent performance of the method compared to the RS-HDMR and Cut-HMDR methods.

In this study, we consider new finite difference schemes for solving the Helmholtz equation. Novel difference schemes which do not introduce truncation error are presented, consequently the exact solution for the Helmholtz equation can be computed numerically. The most important features of the new schemes are that while the resulting linear system has the same simple structure as those derived from the standard central difference method, the technique is capable of solving Helmholtz equation at any wavenumber without using a fine mesh. The proof of the uniqueness for the discretized Helmholtz equation is reported. The power of this technique is illustrated by comparing numerical solutions for solving one- and two-dimensional Helmholtz equations using the standard second-order central finite difference and the novel finite difference schemes.