We study substructuring preconditioners for the linear system arising from the discretization of parabolic problems when the mortar method is applied. By using a suitable non standard norm equivalence we build an efficient edge block preconditioner and we prove a polylogarithmic bound for the condition number of the preconditioned matrix.

In many earth science problems, the scales of interest range from centimeters to kilometers. Computer power and time limitations prevent inclusion of all the fine-scale features in most models. However, upscaling methods allow creation of physically realistic and computationally feasible models. Instead of solving the problem completely on the fine scale, upscaling methods produce a coarse-scale solution that includes some of the fine-scale detail. Operator-based upscaling applied to the pressure/acceleration formulation of the acoustic wave equation solves the problem via decomposition of the solution into coarse and subgrid pieces. To capture local fine-scale information, small subgrid problems are solved independently in each coarse block. Then these local subgrid solutions are included in the definition of the coarse problem. In this paper, accuracy of the upscaled solution is determined via a detailed finite element analysis of the continuous-in-time and fully-discrete two-scale numerical schemes. We use lowest-order Raviart-Thomas mixed finite element approximation spaces on both the coarse and fine scales. Energy techniques show that in the $L^2$ norm the upscaled acceleration converges linearly on the coarse scale, and pressure (which is not upscaled in this implementation) converges linearly on the fine scale. The fully discrete scheme is also shown to be second-order in time. Three numerical experiments confirm the theoretical rate of convergence results.

Based on some coupled discretizations, a local computational scheme is proposed and analyzed in this paper for a class of higher order finite element eigenvalue approximations. Its efficiency is proven by theoretical and numerical evidences. It is shown that the solution of an eigenvalue problem in a higher order finite element space may be reduced to the solution of an eigenvalue problem in a lower order finite element space, and the solutions of some linear algebraic systems in the higher order finite element space by some local and parallel procedure.

This paper studies the problem of controller design for networked control systems regulated by a network data transmission protocol proposed in [50]. In this framework, the plant is first formulated as a mixed logical dynamical (MLD) system, then model predictive control (MPC) based on the mixed-integer programming is adopted to design a controller to guarantee certain control performance. It is shown that the solvability of the finite-horizon MPC is not equivalent to that of the infinite-horizon MPC, which is normally true for most existing MPC methods. The non-convexity feature of this type of networked control systems rules out explicit piecewise affine controllers that are designable for linear convex control systems. Notwithstanding these difficulties, controller design is still feasible due to the special nature of the data transmission strategy, i.e., only a small number of logic values are involved. Furthermore, control of higher-order systems and tracking of more complicated signals can be readily dealt with using this new approach. Two examples are presented to illustrate the strength of the proposed approach.

In this paper, we discuss the boundary value problem for the linear elastic equations in a perforated domain $\Omega^{\varepsilon}$. We fill all holes with a very compliant material, then we study the homogenization method and the multiscale analysis for the associated multiphase problem in a domain $\Omega$ without holes. We are interested in the asymptotic behavior of the solution for the multiphase problem as the material properties of one weak phase go to zero, which has a wide range of applications in shape optimization and in 3-D mesh generation. The main contribution obtained in this paper is to give a full mathematical justification for this limiting process in general senses. Finally, some numerical results are presented, which support strongly the theoretical results of this paper.

In this paper, we solve a problem describing the mechanical deformations of a porous medium in the presence of a monophasic linear flow or a two phase nonlinear flow with the purpose of modelizing subsidence of hydrocarbon reservoirs. An essential characteristics of this problem is that the mechanical deformation and the flow have different time scales. In petroleum industry, one uses different very efficient simulators for the flow problem and the mechanical deformations, which enables to handle complex models. Therefore it is necessary to be able to combine as efficiently as possible the exploitation of these simulators. We propose two alternative splitting approaches. The first one is the staggered algorithm used by engineers, which amounts to a Gauss-Seidel method in the one phase linear case. The second approach is based upon the preconditioned conjugate gradient method. We use a numerical multi-scale method in both of these algorithms. We compare these two approaches and we show that the preconditioned conjugate gradient algorithm is faster and more robust than the staggered algorithm. Applying the preconditioned conjugate gradient algorithm therefore seems to compensate for the fact that the inf-sup condition for the mixed discretization method is not satisfied when combining the simulators for the mechanical deformations and for the flow computations.

Numerical approximations are considered for a mathematical model for miscible displacement influenced by mobile and immobile water. A mixed finite element method is adopted to give a direct approximation of the velocity, the concentration in mobile water is approximated by an alternating direction Galerkin finite element method combined with the method of characteristics and the concentration in immobile water is approximated by a standard Galerkin method. Optimal order $L^2$- and $H^1$-error estimates are derived.

In this paper, we propose a fully drift-implicit splitting numerical scheme for the stochastic differential equations driven by the standard $d$-dimensional Brownian motion. We prove that its strong convergence rate is of the same order as the standard Euler-Maruyama method. Some numerical experiments are also carried out to demonstrate this property. This scheme allows us to use the latest information inside each iteration in the Euler-Maruyama method so that better approximate solutions could be obtained than the standard approach.

This paper investigates construction and approximation properties of the locally divergence-free (LDF) finite elements. Numerical stability of the natural and normalized bases for the LDF elements is analyzed. Error estimates about the jumps and the total divergence of the localized $L_2$-projection are proved and validated through numerical examples.

We propose a pre-processing mesh re-distribution algorithm based upon harmonic maps employed in conjunction with discontinuous Galerkin approximations of advection-diffusion-reaction problems. Extensive two-dimensional numerical experiments with different choices of monitor functions, including monitor functions derived from goal-oriented a posteriori error indicators are presented. The examples presented clearly demonstrate the capabilities and the benefits of combining our pre-processing mesh movement algorithm with both uniform, as well as, adaptive isotropic and anisotropic mesh refinement.

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the $H^1$-approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the $L^2$-stability analysis of the scheme and derive the $L^2$-error approximations.