a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media abstract we develop a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media by utilizing the intrinsic mathematical numerical and physical properties of the mathematical model we model the process in which co2 is injected into a deep aquifer spreads within the aquifer and upon reaching a leaky well rises up to a shallower aquifer to quantify the leakage rate which depends on the pressure build up in the aquifer due to injection and the buoyancy of co2 the underlying eulerian lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in co2 sequestration the sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre existing eulerian lagrangian methods in a nonintrusive manner it also provides a scalable framework to consider uncertainty in a straightforward parallel manner preliminary numerical experiments show the feasibility and potential of the method

necessary and sufficient conditions for the robustness of time varying systems with applications to sampled data systems necessary and sufficient conditions for robustness of time varying systems with structured norm bounded uncertainty are provided the l sup infinity signal norm is used and the conditions are given in terms of the spectral radii of certain matrices obtained from the kernel representation of the time varying system at hand the derived conditions are then applied to continuous time linear time invariant systems and for sampled data systems

analysis of the lattice boltzmann treatment of hydrodynamics it is shown that the lattice boltzmann model can be made to obey the three dimensional thermal navier stokes equations by fixing 26 moments of the equilibrium distribution function this can be achieved on a cubic lattice with particle velocities connecting each lattice site to its first second and third nearest neighbors however this minimal model is not linearly stable in the limit of low transport coefficients additional particle velocities must be selected to achieve linear stability

an optimal order error estimate for the mass conservative characteristic finite element scheme abstract the mass conservative characteristic mcc finite element scheme preserves the mass balance which is an important property in the numerical simulation of the convection diffusion problems in this paper an optimal order error estimate is obtained for the model problem in one space dimension the convergence rate is o h 2 t in the l l 2 norm for the linear finite element some numerical examples are given to confirm the conclusion

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

convergence of a fully conservative volume corrected characteristic method for transport problems we consider the convergence of a volume corrected characteristics mixed method vccmm for advection diffusion systems it is known that without volume correction the method is first order convergent provided there is a nondegenerate diffusion term we consider the advective part of the system and give some properties of the weak solution with these properties we prove that the volume corrected method with no diffusion term gives a lower order l 1 convergence rate of mathcal o h sqrt delta t h delta t r where r is related to the accuracy of the characteristic tracing this result compares favorably to godunov s method but avoids the cfl constraint so large time steps can be taken in practice in fact godunov s method converges at mathcal o h 1 2 which is our result for delta t ch where now c is not limited however the optimal choice delta t ch 2 2r 1 gives a better rate mathcal o h 2r 2r 1 than godunov s method e g mathcal o h 2 3 if r 1 with a nondegenerate diffusion term we obtain an l 2 error estimate for the problem we also prove the existence of and give an error estimate for a perturbed velocity field for which the volume is locally conserved finally some convergence tests are given to verify the optimal convergence rate for r 1

necessary and sufficient conditions of stability a multiloop generalization of the circle criterion a linear time invariant system with a vector output and a vector input is described this system is closed by an uncertain nonlinear time varying feedback the only information on this feedback is given by several integral quadratic inequalities i e the uncertainties under consideration generalize the so called conic nonlinearities necessary and sufficient frequency domain conditions of stability are obtained an advanced version of the s procedure losslessness theorem and some other tools of the absolute stability theory are used

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part ii fully discretized scheme and numerical results we analyze a second order pure lagrange galerkin method for variable coefficient convection diffusion possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions in a previous paper the proposed second order pure lagrangian time discretization scheme was introduced and analyzed for the same problem more precisely the l infty h 1 stability and l infty h 1 error estimates of order o delta t 2 had been obtained moreover for the particular case of incompressible flows stability inequalities with constants independent of the final time have been stated in the present paper l infty h 1 error estimates of order o delta t 2 o h k are obtained for the fully discretized pure lagrange galerkin method to prove these results we use some properties obtained in the previous paper finally numerical tests are presented that confirm the theoretical results

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

a fully mass and volume conserving implementation of a characteristic method for transport problems the characteristics mixed method considers the transport not of a single point or fluid particle but rather the mass in an entire region of fluid this mass is transported along the characteristic curves of the hyperbolic part of the transport equation and the scheme thereby produces very little numerical dispersion conserves mass locally and can use long time steps however since the shape of a characteristic trace back region must be approximated in numerical implementation its volume may be incorrect resulting in inaccurate concentration densities and further inaccurate reaction dynamics we present a simple modification to the characteristics mixed method that conserves both mass and volume of the transported fluid regions our algorithm also handles boundary conditions through a space time change of variables in the trace back routines which allows the boundary to be treated as if it were interior to the domain nearly point sources such as wells present special difficulties since characteristic trace back curves converge in their vicinity we also present techniques that allow one to conservatively implement wells the techniques are illustrated in five numerical examples

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

nonlinear time varying controllers for robust stabilization in this paper we consider the problem of robust stabilizallon for a certain class of plants with unstructured infinite dimensional uncertainty we demonstrate that for the problem of robustly stabilizing this class of plants linear time invariant controllers perform as well as nonlinear time varying nltv ones ihis in particular implies that adaptive control laws offer no advantage as far as the problem of robust stabilization of this class of plants is concerned as a corollary we demonstrate that the small gain theorem is both necessary and sufficient for a certain class of nltv operators

the lowest order characteristic mixed finite element scheme for convection dominated diffusion problem a new characteristic mixed finite element method mfem is proposed by combining the method of characteristics with the new mixed variational formulation for the convection dominated diffusion problem the error estimates for both the original variable u and auxiliary variable p are derived via the properties of the integral identity and mean value techniques at the same time some numerical experiments are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method

a finite element semi lagrangian method with l2 interpolation summary high order accurate methods for convection dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy the state of the art in this field is more advanced for eulerian methods than for semi lagrangian slag methods in this paper we introduce a new slag method that is based on combining the modified method of characteristics with a high order interpolating procedure the method employs the finite element method on triangular meshes for the spatial discretization an l2 interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes numerical results are illustrated for a linear advection diffusion equation with known analytical solution and for the viscous burgers equation the computed results support our expectations for a robust and highly accurate finite element slag method copyright 2012 john wiley sons ltd

a mass conservative characteristic fe scheme for optimal control problems governed by convection diffusion equations abstract in this paper we develop a mass conservative characteristic finite element scheme for optimal control problems governed by linear singularly perturbed convection diffusion equations we discuss the case that the velocity fields are non divergence free the space discretization of the state variable is done by piecewise linear continuous functions whereas the control variable is approximated by piecewise constant functions due to the limitation of the regularity the scheme preserves the mass balance for the original state equation we derive a priori error estimates for both the control and state approximations some numerical examples are presented to show the efficiency of the proposed scheme

heat transfer in lattice bgk modeled fluid the thermal lattice bgk model is a recently suggested numerical tool aiming at solving problems of thermohydrodynamics the quality of the lattice bgk simulation is checked in this paper by calculating temperature profiles in the couette flow under different eckert and mach numbers a revised lower order model is proposed to improve the accuracy and the higher order model is proved to be advantageous in this respect especially in the flow regime with a higher mach number

uniform error analysis for lagrange galerkin approximations of convection dominated problems in this paper we present a rigorous error analysis for the lagrange galerkin method applied to convection dominated diffusion problems we prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit this is in contrast to other results in this field for the time discretization uniform convergence with respect to the diffusion parameter of order o k tn is shown for initial values in l2 and o k for initial values in h2 for the spatial discretization with linear finite elements we verify uniform convergence of order o h2 min h h2 k for data in h2 by interpolation of banach spaces suboptimal convergence rates are derived under less restrictive assumptions the analysis is heavily based on a priori estimates uniform in the diffusion parameter for the solution of the continuous and the semidiscrete problem they are derived in a lagrangian framework by transforming the eulerian coordinates completely into subcharacteristic coordinates finally we illustrate the error estimates by some numerical results

the modified method of characteristics with adjusted advection the mmoc procedure for approximating the solutions of transport dominated diffusion problems does not automatically preserve integral conservation laws leading to mass balance errors in many kinds of flow problems the variant called the mmocaa discussed herein preserves the conservation law at a minor additional computational cost it is shown that its solution in either galerkin or finite difference form converges at the same rates as were proved earlier by dougl as and russell for the standard mmoc procedure

the necessity of the small gain theorem for time varying and nonlinear systems it is shown that an appropriately modified version of the small gain theorem which uses the asymptotic gain of a fading memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators this modified small gain theorem is then used to obtain conditions for robust finite energy input output stability of 1 nonlinear plants subject to nonlinear time varying perturbations and 2 linear time varying plants subject to linear time varying perturbations these conditions are shown to be necessary as well as sufficient

two grid methods for characteristic finite volume element solution of semilinear convection diffusion equations abstract two grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection dominated diffusion equations the methods are based on the method of characteristics two grid method and the finite volume element method the nonsymmetric and nonlinear iterations are only executed on the coarse grid with grid size h and the fine grid solution with grid size h can be obtained by a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 3

uniform estimates for eulerian lagrangian methods for singularly perturbed time dependent problems we prove a priori optimal order error estimates in a weighted energy norm for several eulerian lagrangian methods for singularly perturbed time dependent convection diffusion equations with full regularity the estimates depend only on certain sobolev norms of the initial and right hand side data but not on varepsilon or any norm of the true solution and so hold uniformly with respect to varepsilon we use the interpolation of spaces and stability estimates to derive an varepsilon uniform estimate for problems with minimal or intermediate regularity where the convergence rates are proportional to certain besov norms of the initial and right hand side data

solution of nonlinear convection diffusion problems by a conservative galerkin characteristics method we consider a scheme for nonlinear degenerate convection dominant diffusion problems that arise in contaminant transport in porous media with equilibrium adsorption isotherm this scheme is based on a regularization relaxation scheme that has been introduced by jager and kacur numer math 60 407 427 1991 m2an math model numer anal 29 n5 605 627 1995 with a type of numerical integration by bermejo siam j numer anal 32 425 455 1995 to the modified method of characteristics with adjusted advection mmocaa that was recently developed by douglas et al numer math 83 3 353 369 1999 comput geosci 1 155 190 1997 we present another variant of adjusting advection method the convergence of the scheme is proved an error estimate of the approximated scheme is derived computational experiments are carried out to illustrate the capability of the scheme to conserve the mass

optimization based mesh correction with volume and convexity constraints we consider the problem of finding a mesh such that 1 it is the closest with respect to a suitable metric to a given source mesh having the same connectivity and 2 the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh this volume correction problem arises in important simulation contexts such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi lagrangian transport schemes in this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective while the prescribed cell volumes mesh validity and or cell convexity specify the constraints we solve this problem numerically using a sequential quadratic programming sqp method whose performance scales with the mesh size to achieve scalable performance we develop a specialized multigrid based preconditioner for optimality systems that arise in the application of the sqp method to the volume correction problem numerical examples illustrate the importance of volume correction and showcase the accuracy robustness and scalability of our approach

error analysis for characteristics based methods for degenerate parabolic problems we consider characteristics based finite element methods for solving nonlinear degenerate advection diffusion equations these equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems standard finite element galerkin methods have been studied for these equations in this paper we analyze the characteristics based finite element methods for them the main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity here we develop a technique that respects the degeneracy and the known minimal regularity this technique is based on the green operator for standard elliptic equations and is developed directly for the degenerate advection diffusion equations we concentrate our analysis on the modified method of characteristics mmoc and one of its variants the modified method of characteristics with adjusted advection mmocaa which conserves mass we derive error estimates in various norms the extension to other variants is discussed the present technique is also applied to nondegenerate problems error estimates previously obtained for the mmoc are derived under much weaker regularity assumptions on the solution and the error estimates for the mmocaa appear new even in the nondegenerate case finally numerical results are presented to show the sharpness of the error estimates derived

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

controller design for plants with structured uncertainty this paper addresses the problem of designing feedback controllers to achieve good performance in the presence of structured plant uncertainty and bounded but unknown disturbances a general formulation for the performance robustness problem is presented and exact computable conditions are furnished these conditions are then utilized for synthesizing robust controllers which involves solving l1 optimization problems these solutions are computed using the duality theory of lagrange multipliers approximations and computational issues are discussed

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

weakly compressible lattice boltzmann simulations of reacting flows with detailed thermo chemical models abstract given the complex geometries usually found in practical applications the lattice boltzmann lb method is becoming increasingly attractive for flow simulations in addition to the simple treatment of intricate geometrical configurations lb solvers can be implemented on very large parallel clusters with excellent scalability however reacting flows lead to additional challenges and have seldom been studied by lb methods in this study an in house low mach number lattice boltzmann solver alborz has been extended to take into account multiple chemical components and reactions for this purpose the temperature and species of each mass fraction field are modeled through separate distribution functions the flow distribution function is assumed to be independent of temperature and species mass fractions which is valid in the limit of weak density variations in order to compute reaction terms as well as variable thermodynamic and transport properties the lb code has been coupled to another library of our group regath in this manner lb simulations with detailed chemical kinetics and thermo chemical models become possible since the code is currently limited to weak density variation its performance has been checked for a laminar premixed as well as non premixed counter flow ozone air reacting flow describing kinetics with 4 species and 18 elementary reactions comparisons of the obtained reacting flow structures with results from classical finite difference simulations show excellent agreement

a two grid mmoc finite element method for nonlinear variable order time fractional mobile immobile advection diffusion equations abstract a fully discrete two grid modified method of characteristics mmoc scheme is proposed for nonlinear variable order time fractional advection diffusion equations in two space dimensions the mmoc is used to handle the advection dominated transport and the two grid method is designed for efficiently solving the resulting nonlinear system optimal l 2 error estimates are derived for both the mmoc scheme and the corresponding two grid mmoc scheme numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

an efficient characteristic finite difference s ddm scheme for convection diffusion equations abstract in this paper an efficient splitting domain decomposition method scheme for solving time dependent convection diffusion reaction equations is analyzed a three step method along each direction is used to solve the solution over each block divided sub domain at every time interval the new solutions are firstly solved by the quadratic interpolation then the intermediate fluxes on the interfaces of sub domains are computed by local multi point weighted average from the new solutions lastly the solutions and fluxes in the interiors of sub domains are computed by the splitting implicit characteristic difference method by some auxiliary lemmas and the defined intermediate exact solution the stability and error estimate are given in discrete l 2 norm we further prove that our scheme is of second order convergence in space and of first order convergence in time numerical experiments are presented to validate theoretical result

an enhanced lattice bhatnagar gross krook method for boussinesq flow abstract we derive by the established method of chapman enskog expansion a simple thermal lattice bhatnagar gross krook fluid algorithm with adjustable prandtl number specifically adapted to the simulation of flows in the boussinesq approximation where there is negligible coupling between internal and translational kinetic energy we consider implementation aspects of the model and apply it to the solution of flow in a uniformly heated duct and find the results of our simulations to be in good agreement with theory

an optimal order error estimate to ellam schemes for transient advection diffusion equations on unstructured meshes the eulerian lagrangian localized adjoint method ellam provides a general characteristic procedure for solving transient advection diffusion equations with general boundary conditions in a mass conservative manner in this paper we prove an optimal order error estimate in the l 2 norm and a superconvergence estimate in the energy norm for the ellam scheme to d dimensional transient advection diffusion equations with general flux boundary conditions on unstructured meshes

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

fading memory feedback systems and robust stability abstract this paper considers fading memory for nonlinear time varying systems and associated problems of robust stability we define two notions of fading memory for stable dynamical systems uniform and pointwise we then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory in particular we show that 1 all stable discrete time linear time varying ltv systems have uniform fading memory 2 all stable continuous time ltv systems have pointwise fading memory and 3 stable finite dimensional continuous time ltv systems have uniform fading memory we then show that a version of the small gain theorem which employs the asymptotic gain of a fading memory system is necessary for the stable invertibility of certain feedback operators these results are presented for both continuous time and discrete time systems using general lp or l p notions of input output stability and generalize existing results for l2 stability we further investigate fading memory in a closed loop context for linear plants we parametrize all nonlinear controllers which lead to closed loop pointwise fading memory

uniform estimates of an eulerian lagrangian method for time dependent convection diffusion equations in multiple space dimensions we prove a priori error estimates for a family of eulerian lagrangian methods for time dependent convection diffusion equations in multiple space dimensions which hold uniformly with respect to the vanishing parameter varepsilon we use the interpolation of spaces and stability estimates to derive an varepsilon uniform estimate for problems with minimal or intermediate regularity where the convergence rates are proportional to certain besov norms of the initial and right hand side data these estimates theoretically justify the strength of eulerian lagrangian methods

a characteristics mixed covolume method for a convection dominated transport problem in this paper we propose a characteristics mixed covolume method for approximating the solution to a convection dominated transport problem the method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term the velocity and press are approximated by the lowest order raviart thomas mixed finite element space on rectangles the projection of a mixed covolume element is introduced we prove its first order optimal rate of convergence for the approximate velocities in the l 2 norm as well as for the approximate pressures in the l 2 norm

a new mcc mfe method for compressible miscible displacement in porous media this paper presents a new mass conservative finite element method for solving the compressible miscible displacement problem in porous media this new method combines characteristic tracking with the mixed finite element method specifically the mass conservative characteristic finite element method is applied to the concentration equation whereas the mixed finite element method with continuous flux approximation is used for the flow equation this procedure conserves mass globally convergence of the fully discrete scheme is analyzed and optimal l 2 norm and h 1 norm error estimates for the concentration are obtained numerical experiments are presented to illustrate the theoretical analysis

lattice boltzmann computational fluid dynamics in three dimensions the recent development of the lattice gas method and its extension to the lattice boltzmann method have provided new computational schemes for fluid dynamics both methods are fully paralleled and can easily model many different physical problems including flows with complicated boundary conditions in this paper basic principles of a lattice boltzmann computational method are described and applied to several three dimensional benchmark problems in most previous lattice gas and lattice boltzmann methods a face centered hyper cubic lattice in four dimensional space was used to obtain an isotropic stress tensor to conserve computer memory we develop a model which requires 14 moving directions instead of the usual 24 directions lattice boltzmann models describing two phase fluid flows and magnetohydrodynamics can be developed based on this simpler 14 directional lattice comparisons between three dimensional spectral code results and results using our method are given for simple periodic geometries an important property of the lattice boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency while all other methods including the spectral method are unable to model complicated geometries efficiently

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

a necessary and sufficient condition for robust bibo stability abstract this paper is concerned with obtaining necessary and sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear time varying l stable operators we will show that in some sense the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators

the mmoc and mmocaa schemes for the finite volume element method of convection diffusion problems we present and analyze the modified method of characteristics mmoc and the modified method of characteristics with adjusted advection mmocaa for the finite volume element fve method of convection diffusion problems these two schemes maintain the advantages of both the mmoc and the fve method and the mmocaa scheme discussed herein conserves the conservation law globally at a minor additional computational cost optimal order error estimates in the h 1 norm are proved for these schemes a numerical example is presented to confirm the estimates