Asymptotic Analysis and Optimal Error estimates for Benjamin‐Bona‐Mahony‐Burgers' Type Equations

In this article, stabilization result for the Benjamin‐Bona‐Mahony‐Burgers' (BBM‐B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in , and ‐norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM‐B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in ‐norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.     

An optimal‐order error estimate for a Galerkin‐mixed finite‐element time‐stepping procedure for porous media flows

This article deals with the numerical approximation of miscible displacement problem of one incompressible fluid in a porous medium. The adopted formulation is based on the combined use of a mixed finite‐element scheme to treat pressure equation and of the finite‐element approach to treat concentration equation. Optimal‐order error estimates are obtained under some milder mesh‐parameter constraints. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 707–719, 2012     

Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H¹‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the L^λ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρ_S separated from zero, we establish the uniform‐in‐time H¹‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρ_S separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of time‐splitting schemes for the Stokes problem with stabilized finite elements

The time‐dependent Stokes problem is solved using continuous, piecewise linear finite elements and a classical stabilization procedure. Four order‐one methods are proposed for the time discretization. The first one is nothing but the Euler backward scheme and requires a large linear system involving the velocity and pressure unknowns to be solved. The other three schemes allow velocity and pressure computations to be decoupled, namely the pressure‐matrix method, a method based on an inexact LU factorization, and an operator splitting method. Stability and condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:632–656, 2001     

Global existence,singular solutions,and ill‐posedness for the Muskat problem

The Muskat, or Muskat‐Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele‐Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele‐Shaw problem (the one‐phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher‐viscosity fluid expands into the lower‐viscosity fluid, we show global‐in‐time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher‐viscosity fluid contracts, we construct singular solutions that start off with smooth initial data but develop a point of infinite curvature at finite time. © 2004 Wiley Periodicals, Inc.     

Finite‐time stability and boundedness of switched nonlinear time‐delay systems under state‐dependent switching

This article investigates the finite‐time stability, stabilization, and boundedness problems for switched nonlinear systems with time‐delay. Unlike the existing average dwell‐time technique based on time‐dependent switching strategy, largest region function strategy, that is, state‐dependent switching control strategy is adopted to design the switching signal, which does not require the switching instants to be given in advance. Some sufficient conditions which guarantee finite‐time stable, stabilization, and boundedness of switched nonlinear systems with time‐delay are presented in terms of linear matrix inequalities. Detail proofs are given using multiple Lyapunov‐like functions. A numerical example is given to illustrate the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Complexity 21: 267–275, 2015     

A ray‐integral solution for the wave‐field in the Sommerfeld model

A ray‐integral solution is presented for the wave‐field in the Sommerfeld model (liquid half‐space over solid half‐space), where a point source is placed in the fluid and the two media (fluid and solid) of contrasting densities and wave speeds are homogeneous. This exact closed form solution is then used to evaluate complete time records of the acoustic pressure (at a point receiver located in the fluid in the vicinity of the penetrable fluid‐solid interface) for two Sommerfeld models: one where the shear wave speed in the solid bottom is lower than the sound speed in the fluid and the other where the shear wave speed is higher. These pressure response curves indicate the relative importance of the various wave‐forms (the critically refracted longitudinal and shear waves, the pseudo‐Rayleigh and Stoneley interface waves, the direct wave, and the totally reflected wave) contributing to the solution and the possibility of utilizing the arrival times of the refracted and interface waves to determine the bottom rigidity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gliding motion of bacteria on power‐law slime

The present investigation deals with an undulating surface model for the motility of bacteria gliding on a layer of non‐Newtonian slime. The slime being the viscoelastic material is considered as a power‐law fluid. A hydrodynamical model of motility involving an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum is examined. The non‐linear differential equation resulting from the balance of momentum and mass is solved numerically by a finite difference method with an iteration technique. The manner in which the various exponent values of the power‐law flow affect the structure of the boundary layer is delineated. A comparison is made of the power‐law fluid with the Newtonian fluid. For the power‐law fluid with respect to different power‐law exponent values, shear‐thinning and shear‐thickening effects can be observed, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilization of infinite‐dimensional second‐order system by adaptive PI‐controllers

In this paper adaptive stabilization of infinite‐dimensional undamped second‐order systems is considered in the case where the input and output operators are collocated. The systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by an adaptive PI‐controller (proportional plus integral controller). An energy‐like function and a multiplier function are introduced and adaptive stabilization of the linear second‐order systems is analyzed. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐grid stabilization method for solving the steady‐state Navier‐Stokes equations

We formulate a subgrid eddy viscosity method for solving the steady‐state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A FE‐BE coupling for a fluid‐structure interaction problem: Hierarchical a posteriori error estimates

In this article, we establish a hierarchical a posteriori error estimate for a coupling of finite elements and boundary elements for a fluid‐structure interaction problem posed in two and three dimensions. These methods combine boundary elements for the exterior fluid and finite elements for the elastic structure. We consider two weak formulations, a nonsymmetric one and a symmetric one, which are both uniquely solvable. We present the reliability and efficiency of the error estimates. For the two dimensional case, we compute local error indicators which allow us to develop an adaptive mesh refinement strategy on triangles. For the three dimensional case, we use hexahedrons as elements. Numerical experiments underline our theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB^© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem

We consider conforming finite element (FE) approximations of the time‐dependent, incompressible Navier–Stokes problem with inf‐sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence‐free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39–77 for edge‐stabilized FE approximation of the Navier–Stokes problem. Some numerical experiments complement the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1224–1250, 2015     

A two‐grid stabilized mixed finite element method for Darcy‐Forchheimer model

A two‐grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two‐dimensional Darcy‐Forchheimer model. We use the derivative of a smooth function, , to approximate the derivative of in constructing the two‐grid algorithm. The two‐grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two‐grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.