Initial-value problem for a continuous second-order differential equation

We obtain necessary initial conditions for a continuous second-order differential equation.     

Local–global multiscale model reduction for flows in high-contrast heterogeneous media

In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium

In this paper we give a detailed study of the global attractors for parabolic equations governed by the p-Laplacian in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is presented by showing that their ε-Kolmogorov entropy behaves as a polynomial of the variable 1/ε as ε tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ε-entropy of infinite-dimensional attractors are also obtained.     

Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with a certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.     

The long‐time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd.     

Prebreakdown condition in selenium rectifiers

This paper reports data from an experimental investigation of the prebreakdown and breakdown conditions of factory production selenium rectifiers in the temperature range from 100 ° to –196 ° C when a unit voltage pulse with steep leading edge (rise time about 10^–8 sec) is applied to the rectifier in the non-conducting direction. Change in voltage across the rectifier and in current through it were recorded by a two-channel high-speed electronic oscilloscope. On the basis of this test data, the variation of discharge rise-time in the selenium p-n junction as a function of overvoltage and temperature was determined, the formation of current pulses was detected in the junction, both before and during breakdown in the temperature range from 100 ° to –196 ° C, and it was also established that breakdowns of selenium rectifier p-n junctions are due to joint action of the Zener effect and impact ionization.We are indebted to Prof. Kh. I. Amirkhanov for his constant interest in this work.     

Simultaneous oscillation of several independently tunable spectral lines in a distributed feedback dye laser

A simple method is proposed which allows obtaining distributed-feedback laser action simultaneously at several spectral lines of any wavelength with independent tuning within the gain profile of the dye.