dynamic scaling in miscible viscous fingering we consider dynamic scaling in gravity driven miscible viscous fingering we prove rigorous one sided bounds on bulk transport and coarsening in regimes of physical interest the analysis relies on comparison with solutions to one dimensional conservation laws and new scale invariant estimates our bounds on the size of the mixing layer are of two kinds a naive bound that is sharp in the absence of diffusion and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion the naive bound is simple and robust but does not yield the experimental speed of transport in a reduced model derived by wooding 20 we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments wooding s model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

characteristic local grid refinement techniques for reservoir flow problems using a sequential time marching approach we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium by combining adaptive grid refinement with an operator splitting technique based on the modified method of characteristics the saturation equation is treated in a consistent and accurate way for the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

convergence estimates for semidiscrete parabolic equation approximations semidiscrete methods for approximating the solutions of initial boundary value problems for parabolic equations are studied the construction of these semidiscrete methods is based upon the availability of several different galerkin type approximation methods for the associated elliptic steadystate problem the properties required of the spacial discretization methods are listed and estimates of the error made by the resulting semidiscrete approximations and of their time derivatives are given in particular estimates are given that require only weak smoothness assumptions on the initial data verifications of the required properties for various galerkin type methods are provided

adaptive digital control of hammerstein nonlinear systems with limited output sampling this paper is motivated by the practical control considerations that nonlinearity is abundant in industrial processes and output sampling rates are often limited due to hardware constraints in particular for a hammerstein nonlinear sampled data system in which the output sampling period is an integer multiple of the input updating period we derive by using a polynomial transformation technique a mathematical model which is suitable for parameter estimation with dual rate measurement data further we present an adaptive control scheme for such a dual rate nonlinear system the parameter estimation based adaptive algorithm can achieve virtually asymptotically optimal control and ensure that the closed loop system is stable and globally convergent the simulation results are included

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

modeling studies and efficient numerical methods for proton exchange membrane fuel cell abstract in this paper a three dimensional 3d nonisothermal multiphysics two phase steady state transport model and its efficient numerical methods are systematically studied for a full proton exchange membrane fuel cell pemfc in the sense of efficiency and accuracy the conservation equations of mass momentum species charge and energy are fully addressed in view of nonisothermality and multiphase characteristics in addition from an accurate numerical discretization s point of view we present some new formulations for species equations by investigating the interactions among the species in a framework of the combined finite element upwind finite volume method some efficient numerical methods are developed in terms of kirchhoff transformation for the sake of a fast and convergent numerical simulation the 3d simulations demonstrate that the convergent solutions can be attained within 80 nonlinear iterations in contrast to the oscillating and nonconvergent iterations conducted by commercial flow solvers or in house code with standard finite element volume methods numerical convergence tests are carried out to verify the efficiency and accuracy of our numerical algorithms and techniques

convergence of implicit finite difference methods applied to nonlinear mixed systems a new technique to prove convergence of finite difference methods applied to nonlinear pdes arising in computational fluid dynamics is presented the underlying systems may be hyperbolic parabolic or of mixed type like the navier stokes equations implicit finite difference methods are analyzed the essential idea leading to success is the introduction of a pilot function that is highly attractive to the numerical approximation and converges itself to the solution of the underlying system

analysis of fully discrete fem for miscible displacement in porous media with bear scheidegger diffusion tensor fully discrete galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly used bear scheidegger diffusion dispersion tensor begin aligned d mathbf u gamma d m i mathbf u bigg alpha t i alpha l alpha t frac mathbf u otimes mathbf u mathbf u 2 bigg end aligned previous works on optimal order l infty 0 t l 2 norm error estimate required the regularity assumption nabla x partial td mathbf u x t in l infty 0 t l infty omega while the bear scheidegger diffusion dispersion tensor is only lipschitz continuous even for a smooth velocity field mathbf u in terms of the maximal l p regularity of fully discrete finite element solutions of parabolic equations optimal error estimate in l p 0 t l q norm and almost optimal error estimate in l infty 0 t l q norm are established under the assumption of d mathbf u being lipschitz continuous with respect to mathbf u

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

numerical approximations for a miscible displacement process in porous media a reservoir simulation problem describing a miscible displacement process is stated and analyzed and a semidiscrete numerical scheme based on a galerkin method with mesh refinements is presented for its solution the underlying physical problem involves two miscible incompressible fluids of equal viscosity flowing in a porous medium injection and production wells are rigorously modelled which leads to a formulation containing nearly singular source and sink terms the singularities arising in the solution are described in some detail and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements

on existence and uniqueness results for a coupled system modeling miscible displacement in porous media initial boundary value problems for a nonlinear differential system of two equations are considered in this paper this system of equations one of elliptic form for the pressure and the other of parabolic form for the concentration of one fluid governs the miscible displacement of one incompressible fluid by another in a porous medium under some reasonable assumptions on the data it is proved that the system possesses a weak solution and uniqueness is also established for the semiclassical solution of the system

implementation of the entropy viscosity method with the discontinuous galerkin method the notion of entropy viscosity method introduced in guermond and pasquetti 21 is extended to the discontinuous galerkin framework for scalar conservation laws and the compressible euler equations

linearized fe approximations to a nonlinear gradient flow we study fully discrete linearized galerkin finite element approximations to a nonlinear gradient flow applications of which can be found in many areas due to the strong nonlinearity of the equation existing analyses for implicit schemes require certain restrictions on the time step and no analysis has been explored for linearized schemes this paper focuses on the unconditionally optimal l 2 error estimate of a linearized scheme the key to our analysis is an iterated sequence of time discrete elliptic equations and a rigorous analysis of its solution we prove the w 1 infty boundedness of the solution of the time discrete system and the corresponding finite element solution based on a more precise estimate of elliptic pdes in w 2 2 epsilon 1 and h 2 epsilon 2 and a physical feature of the gradient dependent diffusion coefficient numerical examples are provided to support our theoretical analysis

a finite difference scheme for some nonlinear diffusion equations in an absorbing medium support splitting phenomena in the porous media equation vt vm xx m 1 it is well known that there appears a finite propagation of the initial support and that if the initial support is connected so is mbox supp v t cdot for t 0 when the effect of the absorption is considered as the additional lower order term cvp c 0 p 0 the possibility that the support will split is expected rosenau and kamin phys d 8 1983 pp 273 283 tried the numerical computations and suggested the support splitting phenomena but the theoretical justification is not discussed in this paper such phenomena are investigated by use of finite difference schemes and the sufficient conditions under which the support begins to split are obtained in the specific case where m p 2 and 0 p 1

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

droplet spreading intermediate scaling law by pde methods we consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity the standard lubrication approximation leads to an evolution equation for the film height h that is ill posed when the spreading is limited by the no slip boundary condition at the liquid solid interface due to a singularity at the moving contact line the most common relaxation of the no slip boundary condition removes this singularity but introduces a new physical length scale the slippage length b it is believed that this microscopic length scale only enters logarithmically in the effective that is macroscopic spreading behavior in this paper we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b more precisely we prove a scaling law for the diameter of the apparent that is macroscopic support of the droplet in time this is an intermediate scaling law it takes an initial layer to forget the initial droplet shape whereas after a long time the droplet is so thin that its spreading is governed by the physics on the scale b our proof works by deriving suitable estimates for physically relevant integral quantities the free energy the length of the apparent support and their respective rates of change as opposed to matched asymptotic methods this pde approach closely mimics a simple heuristic argument based on the gradient flow structure 2002 john wiley sons inc

identification of hammerstein nonlinear armax systems two identification algorithms an iterative least squares and a recursive least squares are developed for hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by armax carma models the basic idea is to replace unmeasurable noise terms in the information vectors by their estimates and to compute the noise estimates based on the obtained parameter estimates convergence properties of the recursive algorithm in the stochastic framework show that the parameter estimation error consistently converges to zero under the generalized persistent excitation condition the simulation results validate the algorithms proposed

stability condition for difference schemes for parabolic systems recently goldberg studied in m goldberg siam j numer anal 35 1998 pp 478 493 m goldberg siam j numer anal 35 1998 pp 1995 2003 stability conditions of a well known family of difference schemes for the following multidimensional parabolic system of second order frac partial bu bf x t partial t sum 1 le p le q le s a pq frac partial 2 bu bf x t partial x p partial x q sum 1 le p le s b p frac partial bu bf x t partial x p c bu bf x t here we present a modified stability condition for the above problem and prove that this condition is optimal in some sense

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

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

circle criteria in recursive identification positive real conditions and differential sector conditions have recently been shown to imply global convergence w p 1 for recursive identification schemes based on a class of single input single output linear wiener models the models consist of linear dynamics followed by a static output nonlinearity the model structure is hence closely related to that of the lure problem in the stability theory of feedback systems this paper proves that the conditions for convergence can be transformed to graphical circle criteria depending on the sector conditions and on the nyquist plot of a transfer function related to the prior knowledge of the poles of the identified system

unconditional stability and convergence of crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system the paper is concerned with the unconditional stability and optimal l 2 error estimates of linearized crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system in mathbb r d d 2 3 by introducing a corresponding time discrete system we separate the error into two parts i e the temporal error and the spatial error since the latter is tau independent the uniform boundedness of numerical solutions in l infty norm follows an inverse inequality immediately without any restrictions on time stepsize then optimal error estimates are obtained in a routine way numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

discontinuous galerkin finite element convergence for incompressible miscible displacement problems of low regularity in this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous galerkin method under minimal regularity assumptions the main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem in order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries the careful construction of a reflexive subspace of the space of bounded variation which compactly embeds into l 2 omega and of a lifting operator which is compatible with the nonlinear diffusion coefficient are required an equivalent skew symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method numerical experiments underline the robustness of the proposed algorithm

block monotone iterative method for semilinear parabolic equations with nonlinear boundary conditions two block monotone iterative schemes called jacobi and gauss seidel monotone iterations are presented for numerical solutions of a class of semilinear parabolic equations under nonlinear boundary conditions by the finite difference method these iteration schemes extend the method for semilinear elliptic boundary value problems to parabolic equations including a comparison result between them it is shown that by using an upper solution and a lower solution as initial iterations each of the iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system some error estimates and a convergence theorem are given and various sufficient conditions for the construction of upper and lower solutions are obtained numerical results are presented for some physical model problems including some problems with known continuous solutions and two problems with l shaped and trapezoidal domains

an efficient s ddm iterative approach for compressible contamination fluid flows in porous media in this paper we develop an efficient splitting domain decomposition method s ddm for compressible contamination fluid flows in porous media over multiple block divided sub domains by combining the non overlapping domain decomposition splitting linearization and extrapolation techniques the proposed s ddm iterative approach divides the large domain into multiple block sub domains in each time interval the s ddm scheme is applied to solve the water head equation in which an efficient local multilevel scheme is used for computing the values of water head on the interfaces of sub domains and the splitting implicit scheme is used for computing the interior values of water head in sub domains and the s ddm scheme is then proposed to solve the concentration equation by combining the upstream volume technique numerical experiments are performed and analyzed to illustrate the efficiency of the s ddm iterative approach for simulating compressible contamination fluid flows in porous media the developed method takes the excellent attractive advantages of both the non overlapping domain decomposition and the splitting technique and reduces computational complexities large memory requirements and long computational durations

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

a new error analysis of characteristics mixed fems for miscible displacement in porous media the method of characteristics type is especially effective for convection dominated diffusion problems due to the nature of characteristic temporal discretization the method allows one to use a large time step in many practical computations while all previous theoretical analyses always required certain restrictions on the time stepsize here we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in mathbb r d d 2 3 for this purpose we introduce a new characteristic time discrete system we prove that the l 2 error bound the characteristic time discrete systemof the fully discrete method of characteristics to the time discrete system is tau independent and the numerical solution is bounded in w 1 infty norm unconditionally with the boundedness optimal error estimates are established in a traditional manner numerical results confirm our th

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

stability and convergence analysis of a decoupled algorithm for a fluid fluid interaction problem in this paper we analyze the stability and error estimate of a decoupled method for a fluid fluid interaction problem this method applies the partitioned time stepping method to decouple the coupling system which is the implicit explicit scheme and was first proposed by connors howell and layton siam j numer anal 50 2012 pp 1297 1319 theoretically we prove that this decoupled algorithm is not only conditional stable but also long time stable we also derive that its error convergence order is o delta t h frac 3 2 for velocity variables in l 2 norm besides several numerical tests are given to verify our theoretical results

anomalous exponents and dipole solutions for the thin film equation we investigate similarity solutions of the thin film equation in particular we look at solutions on the half line x ge0 with compact support and zero contact angle boundary conditions in x 0 such dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind using a combination of phase space analysis and numerical simulations we numerically construct trajectories representing these solutions at the same time obtaining broader insight into the nature of the four dimensional phase space additional asymptotic analysis provides further information concerning the evolution to self similarity

auxiliary model based least squares identification methods for hammerstein output error systems abstract the difficulty in identification of a hammerstein a linear dynamical block following a memoryless nonlinear block nonlinear output error model is that the information vector in the identification model contains unknown variables the noise free true outputs of the system in this paper an auxiliary model based least squares identification algorithm is developed the basic idea is to replace the unknown variables by the output of an auxiliary model convergence analysis of the algorithm indicates that the parameter estimation error consistently converges to zero under a generalized persistent excitation condition the simulation results show the effectiveness of the proposed algorithms

identification of certain time varying nonlinear wiener and hammerstein systems the problem of identification and tracking of time varying nonlinear systems is addressed in particular the wiener system that consists of a dynamic time varying linear part followed by a fixed nonlinearity and the hammerstein system in which the order of these two blocks is reversed are studied the extended kalman filter ekf algorithm is applied it is also shown that this algorithm can be reformulated in terms of a nonlinear minimization problem with a quadratic inequality constraint in order to ensure exponential stability resulting in the algorithm cekf as indicated by means of numerical examples this latter algorithm is less sensitive to the chosen initialization than the ekf the proposed algorithms depend on certain second order statistics that may be unknown in a typical scenario a method for estimation of these quantities is proposed it is demonstrated that the suggested algorithms can be successfully applied to the problem of acoustic echo cancelation

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

output error convergence of adaptive filters with compensation for output nonlinearities output error convergence of a wiener model based nonlinear stochastic gradient algorithm is analyzed the normalized scheme estimates the parameters of a linear finite impulse response model in cascade with a known output nonlinearity the algorithm can be interpreted as a normalized least mean square algorithm with compensation for an output nonlinearity linearizing inversion of the nonlinearity is not utilized global output error convergence is then proved provided that the nonlinearity is monotone not strictly monotone and provided that a previously observed mechanism resulting in deadlock does not occur the algorithm and the analysis include important practical cases like sensor saturation and dead zones that must be excluded when global parametric convergence is studied

a fully coupled upwind discontinuous galerkin method for incompressible porous media flows high order computations of viscous fingering instabilities in complex geometry we present a new approach to the simulation of viscous fingering instabilities in incompressible miscible displacement flows in porous media in the past high resolution computational simulations of viscous fingering instabilities have always been performed using high order finite difference or fourier spectral methods which do not posses the flexibility to compute very complex subsurface geometries our approach instead by means of a fully coupled nonlinear implementation of the discontinuous galerkin method possesses a fundamental differentiating feature in that it maintains high order accuracy on fully unstructured meshes in addition the proposed method shows very low sensitivity to mesh orientation in contrast with classical finite volume approximation used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging computational problems

on source type solutions and the cauchy problem for a doubly degenerate sixth order thin film equation i local oscillatory properties abstract as a key example the sixth order doubly degenerate parabolic equation from thin film theory u t u m u x x x x x n u x x x x x x in r r with two parameters n 0 and m n n 2 is considered in this first part of the research various local properties of its particular travelling wave and source type solutions are studied most complete analytic results on oscillatory structures of these solutions of changing sign are obtained for m 1 by an algebraic geometric approach with extension by continuity for m 1

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

unconditional superconvergence analysis of a new mixed finite element method for nonlinear sobolev equation in this paper a new mixed finite element scheme is proposed for the nonlinear sobolev equation by employing the finite element pair q11 q01 q10 based on the combination of interpolation and projection skill as well as the mean value technique the independent superclose results of the original variable u in h1 norm and the variable q a u u t b u u in l2 norm are deduced for the semi discrete and linearized fully discrete systems is the temporal partition parameter what s more the new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained unconditionally while previous literature always require certain time step conditions finally some numerical results are provided to confirm our theoretical analysis and show the efficiency of the method

a locally conservative finite element method based on piecewise constant enrichment of the continuous galerkin method this paper presents a locally conservative finite element method based on enriching the approximation space of the continuous galerkin method with elementwise constant functions the proposed method has a smaller number of degrees of freedom than the discontinuous galerkin method numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method we also present a theoretical analysis of the method and establish optimal convergence of numerical solutions

unconditionally optimal error estimates of a crank nicolson galerkin method for the nonlinear thermistor equations this paper focuses on unconditionally optimal error analysis of an uncoupled and linearized crank nicolson galerkin finite element method for the time dependent nonlinear thermistor equations in d dimensional space d 2 3 in our analysis we split the error function into two parts one from the spatial discretization and one from the temporal discretization by introducing a corresponding time discrete elliptic system we present a rigorous analysis for the regularity of the solution of the time discrete system and error estimates of the time discretization with these estimates and the proved regularity optimal error estimates of the fully discrete crank nicolson galerkin method are obtained unconditionally numerical results confirm our analysis and show the efficiency of the method

finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials in this paper we study heat and moisture transport through porous textile materials with phase change described by a degenerate nonlinear and strongly coupled parabolic system an uncoupled finite difference method with semi implicit euler scheme in time direction is proposed for the system we prove the existence and uniqueness of the solution of the finite difference system the optimal error estimates in both discrete l 2 and h 1 norms are obtained under the condition that the mesh sizes and h are smaller than a positive constant which depends solely upon physical parameters involved numerical results are presented to confirm our theoretical analysis and compared with experimental data

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

optimal error estimates of linearized crank nicolson galerkin fems for the time dependent ginzburg landau equations in superconductivity in this paper we study linearized crank nicolson galerkin finite element methods for time dependent ginzburg landau equations under the lorentz gauge we present an optimal error estimate for the linearized schemes almost unconditionally i e when the spatial mesh size h and the temporal step tau are smaller than a given constant while previous analyses were given only for some schemes with strong restrictions on the time step size the key to our analysis is the boundedness of the numerical solution in some strong norm we prove the boundedness for the cases tau ge h and tau le h respectively the former is obtained by a simple inequality with which the error functions at a given time level are bounded in terms of their average at two consecutive time levels and the latter follows a traditional way with the induction inverse inequality two numerical examples are investigated to confirm our theoretical analysis and to show clearly that no time step condition is needed

weak solutions of coupled dual porosity flows in fractured rock mass and structured porous media abstract this paper deals with a fully nonlinear degenerate parabolic system with natural critical growths and under non linear boundary conditions such problems arise from the heat and water flow through a partially saturated fractured rock mass and structured porous media existence of a global weak solution of the problem on an arbitrary interval of time is proved by means of semidiscretization in time deriving suitable a priori estimates based on w 1 p regularity of the approximate solution and by passing to the limit from discrete approximations

fuzzy adaptive control of nonlinear uncertain plants with unknown dead zone output this paper focuses on a tracking problem for a class of strict feedback nonlinear systems with unknown dead zone output and unmeasurable states based on the adaptive backstepping technique a new adaptive fuzzy control scheme is proposed it is shown that the proposed control scheme guarantees that all the signals in the closed loop system are semi globally uniformly ultimately bounded the tracking error converges to a small neighborhood of the origin moreover the proposed control method requires only one adaptive law for an nth order system and the structure of the controllers is very simple so they are easy to be implemented the main advantage of this paper is that the unknown dead zone nonlinearity in the output mechanism of strict feedback systems is considered by introducing a nussbaum function the problem of unknown virtual control coefficient is resolved which is caused by the nonlinearity in the output mechanism finally simulation results are provided to show the effectiveness of the proposed approach

convergence analysis of a mixed finite volume scheme for an elliptic parabolic system modeling miscible fluid flows in porous media we study a finite volume discretization of a strongly coupled elliptic parabolic pde system describing miscible displacement in a porous medium we discretize each equation by a finite volume scheme which allows a wide variety of unstructured grids in any space dimension and gives strong enough convergence for handling the nonlinear coupling of the equations we prove the convergence of the scheme as the time and space steps go to 0 finally we provide numerical results to demonstrate the efficiency of the proposed numerical scheme

a meshless method based on point interpolation method pim for the space fractional diffusion equation this paper aims to develop a meshless approach based on the point interpolation method pim for numerical simulation of a space fractional diffusion equation two fully discrete schemes for the one dimensional space fractional diffusion equation are obtained by using the pim and the strong forms of the space diffusion equation numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach

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

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

unconditional optimal error estimates for bdf2 fem for a nonlinear schrodinger equation abstract this paper analyzes unconditional optimal error estimates for a 2 step backward differentiation formula bdf2 method for a nonlinear schrodinger equation in the analysis we split an error estimate into two parts one from the spatial discretization and the other from the temporal discretization we present the boundedness of the solution of the time discrete system in the certain strong norms and the error estimates for time discretization by these boundedness and temporal error estimates we obtain the l 2 error estimates without any conditions on a time step size numerical experiments are provided to validate our analysis and check the efficiency of our method

error estimates of splitting galerkin methods for heat and sweat transport in textile materials this paper is concerned with a single component model of heat and vapor sweat transport through three dimensional porous textile materials with phase change which is described by a nonlinear degenerate and strongly coupled parabolic system an uncoupled splitting galerkin method with semi implicit euler scheme in time direction is proposed for the system in this method a linearized scheme is applied for the approximation to darcy s velocity simultaneously in the mass and energy equations which leads to physical conservation of the method in the flow convection the existence and uniqueness of solution of the finite element system is proved and the optimal error estimate in an energy norm is obtained numerical results are presented to confirm our theoretical analysis and are compared with experimental data

a characteristics mixed finite element method for advection dominated transport problems we define a new finite element method called the characteristics mixed method for approximating the solution to an advection dominated transport problem the method is based on a space time variational form of the advection diffusion equation our test functions are piecewise constant in space and in time they approximately follow the characteristics of the advective i e hyperbolic part of the equation thus the scheme uses a characteristic approximation to handle advection in time this is combined with a low order mixed finite element spatial approximation of the equation boundary conditions are incorporated in a natural and mass conservative fashion the scheme is completely locally conservative in fact on the discrete level fluid is transported along the approximate characteristics a postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux this has the effect of improving the rate of convergence of

a numerical study of air vapor heat transport through textile materials with a moving interface this paper focuses on the numerical study of heat and moisture transfer in clothing assemblies which is described by a multi component and multiphase air vapor heat flow with a moving interface a splitting semi implicit finite volume method is applied for the system of nonlinear parabolic equations and an implicit euler scheme is used for the interface equation in terms of classical dirichlet to neumann map the implicit system can be solved directly and no iteration is needed two types of clothing assemblies are investigated and the comparison with experimental measurements is also presented

compensation of the rls algorithm for output nonlinearities it is shown how the recursive least squares rls algorithm can be modified to compensate for a priori known errors of linearity in the output measurement a novel signal model is used for this purpose only the nonlinear effects are modeled by an output error model and much of the output measurements are used directly in the regression vector the main benefit with this approach is that the advantages of the rls like quick initial convergence for infinite impulse response models can be retained for small linearity errors at the same time the output nonlinearity is allowed to be noninvertible this can be important to treat for example small deadzones and also to avoid the amplification of additive measurement disturbances such amplification can result from inversion of the output nonlinearity simulations illustrate the performance of the algorithm

finite difference methods for two phase incompressible flow in porous media two phase incompressible flow in porous media is governed by a system of nonlinear partial differential equations convection physically dominates diffusion and the object of this paper is to develop a finite difference procedure that reflects this dominance the pressure equation which is elliptic in appearance is discretized by a standard five point difference method the concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms a convergence analysis is given for the method

adaptive enriched galerkin methods for miscible displacement problems with entropy residual stabilization we present a novel approach to the simulation of miscible displacement by employing adaptive enriched galerkin finite element methods eg coupled with entropy residual stabilization for transport in particular numerical simulations of viscous fingering instabilities in heterogeneous porous media and hele shaw cells are illustrated eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions the method provides locally and globally conservative fluxes which are crucial for coupled flow and transport problems moreover eg has fewer degrees of freedom in comparison with discontinuous galerkin dg and an efficient flow solver has been derived which allows for higher order schemes dynamic adaptive mesh refinement is applied in order to reduce computational costs for large scale three dimensional applications in addition entropy residual based stabilization for high order eg transport systems prevents spurious oscillations numerical tests are presented to show the capabilities of eg applied to flow and transport enriched galerkin approximations for coupled flow and transport system entropy residual stabilization for enriched galerkin to stabilize higher order transport scheme dynamic mesh adaptivity for viscous fingering in two and three dimensions

two compressible immiscible fluids in porous media abstract we consider a model of flow of two compressible and immiscible phases in a three dimensional porous media the equations are obtained by the conservation of the mass of each phase this model is treated in its general form with the whole nonlinear terms the only assumption concerns the dependence of densities on a global pressure we obtain the existence of weak solutions under different kinds of degeneracies of the capillary terms

numerical solutions of the incompressible miscible displacement equations in heterogeneous media abstract this paper presents a numerical method based on mixed finite element discontinuous galerkin methods in space and high order runge kutta method in time for solving the miscible displacement problem no slope limiters are needed the proposed method exhibits high order of convergence in space and time when comparing with analytical solutions the simulation shows robustness of the method for heterogeneous media with highly varying permeabilities

on center subspace behavior in thin film equations the large time behavior of weak nonnegative solutions of the thin film equation tfe with absorption u t nabla cdot u n nabla delta u u p 1 u with parameters n in 0 3 and p 1 is studied the standard free boundary problem fbp with zero height zero contact angle and zero flux conditions at the interface and bounded compactly supported initial data is considered it is shown that there exists the critical absorption exponent p 0 1 n frac 4 n such that for p p 0 the asymptotic behavior of solutions u x t for t gg1 is represented by the well known source type solution of the pure tfe absorption u s x t t beta n f y y x t beta with the exponent beta frac 1 4 nn which is perturbed by a couple of ln t factors for n 1 this behavior is associated with the center subspace for the rescaled linearized thin film operator and is given by u x t sim nobreak t ln t beta n f x t beta ln t beta n 4 with beta frac 1 4 n where f y fr

a discontinuous galerkin method for gravity driven viscous fingering instabilities in porous media we present a new approach to the simulation of gravity driven viscous fingering instabilities in porous media flow these instabilities play a very important role during carbon sequestration processes in brine aquifers our approach is based on a nonlinear implementation of the discontinuous galerkin method and possesses a number of key features first the method developed is inherently high order and is therefore well suited to study unstable flow mechanisms secondly it maintains high order accuracy on completely unstructured meshes the combination of these two features makes it a very appealing strategy in simulating the challenging flow patterns and very complex geometries of actual reservoirs and aquifers this article includes an extensive set of verification studies on the stability and accuracy of the method and also features a number of computations with unstructured grids and non standard geometries

numerical entropy production for central schemes in this work a detailed study of the numerical entropy production for both low and high order central schemes is carried out our data show that entropy production can be used to signal the presence of shocks moreover once shocked cells have been identified the spurious entropy production occurring in the remaining cells mimics the behavior of the local error and therefore it can be used as an a posteriori error indicator applications to shock tracking and scheme adaptivity are given we also investigate numerically the entropic properties of third and fourth order central weno schemes to our knowledge this is the first study of the entropic behavior of such high order schemes

stability analysis of explicit entropy viscosity methods for non linear scalar conservation equations this material is based upon work supported by the department of homeland security under agreement 2008 dn 077 ari018 02 national science foundation grants dms 0811041 dms 0914977 dms 1015984 af office of scientific research grant fa99550 12 0358 and is partially supported by award kus c1 016 04 made by king abdullah university of science and technology kaust

an optimal order error estimate for a family of ellam mfem approximations to porous medium flow mathematical models used to describe porous medium flow lead to coupled systems of time dependent nonlinear partial differential equations which present serious mathematical and numerical difficulties standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid orientation effect the ellam mfem time stepping procedure in which an eulerian lagrangian localized adjoint method ellam is used to solve the transport equation and a mixed finite element method mfem is used for the pressure equation simulates porous medium flow accurately even if large spatial grids and time steps are used in this paper we prove an optimal order error estimate for a family of ellam mfem approximations

a finite volume scheme for nonlinear parabolic equations derived from one dimensional local dirichlet problems in this paper we propose a new method to compute the numerical flux of a finite volume scheme used for the approximation of the solution of the nonlinear partial differential equation u t div q f u u 0 in a 1d 2d or 3d domain the function is supposed to be strictly increasing but some values s such that s 0 can exist the method is based on the solution at each interface between two control volumes of the nonlinear elliptic two point boundary value problem q f 0 with dirichlet boundary conditions given by the values of the discrete approximation in both control volumes we prove the existence of a solution to this two point boundary value problem we show that the expression for the numerical flux can be yielded without referring to this solution furthermore we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem numerical results show the increase of accuracy due to the use of this scheme compared to some other schemes

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

mathematical theory of stationary miscible filtration abstract we consider a coupled system of two elliptic partial differential equations that models stationary incompressible two component miscible displacement in porous media we suppose that the dispersion tensor depends on the velocity the so called peaceman model see e g d w peaceman soc pet eng j 6 1966 213 216 assuming that the mobility ratio is not too large we prove the existence of at least one solution in w r 2 w r 2 furthermore supposing that the data are small we also establish well posedness in w r 2 w l 1

well posedness of the cauchy problem for a fourth order thin film equation via regularization approaches abstract this paper is devoted to some aspects of well posedness of the cauchy problem the cp for short for a quasilinear degenerate fourth order parabolic thin film equation the tfe 4 0 1 u t u n u in r n r u x 0 u 0 x in r n where n 0 is a fixed exponent with bounded smooth compactly supported initial data dealing with the cp for at least n 0 3 2 requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces the main goal of the paper is to detect proper solutions of the cp for the degenerate tfe 4 by uniformly parabolic analytic e regularizations at least for values of the parameter n sufficiently close to 0 firstly we study an analytic homotopy approach based on a priori estimates for solutions of uniformly parabolic analytic e regularization problems of the form u t e u u in r n r where e u for e 0 1 is an analytic e regularization of the problem 0 1 such that 0 u u n and 1 u 1 using a more standard classic technique of passing to the limit in integral identities for weak solutions however this argument has been demonstrated to be non conclusive basically due to the lack of a complete optimal estimate regularity theory for these types of problems secondly to resolve that issue more successfully we study a more general similar analytic homotopy transformation in both the parameters as e 0 and n 0 and describe branching of solutions of the tfe 4 from the solutions of the notorious bi harmonic equation u t 2 u in r n r u x 0 u 0 x in r n which describes some qualitative oscillatory properties of cp solutions of 0 1 for small n 0 providing us with the uniqueness of solutions for the problem 0 1 when n is close to 0 finally riemann like problems occurring in a boundary layer construction that occur close to nodal sets of the solutions as e 0 are discussed in other to get uniqueness results for the tfe 4 0 1

unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation galerkin finite element approximation to nonlinear parabolic equation is studied with a linearized backward euler scheme the error between the exact solution and the numerical solution is split into two parts which are called the temporal error and the spatial error through building a time discrete system on one hand the temporal error derived skillfully leads to the regularity of the time discrete system solution on the other hand the independent spatial error and the boundedness of the numerical solution in l norm is deduced with the above achievements at last the superclose result of order o h 2 in h1 norm is obtained without any restriction of in a routine way here h is the subdivision parameter and the time step

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

global existence of weak solution to the heat and moisture transport system in fibrous porous media abstract this paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries which is described by a degenerate and strongly coupled parabolic system we prove the global in time existence of weak solution by constructing an approximate solution with some standard smoothing the proof is based on the physical nature of gas convection in which the heat energy flux in convection is determined by the mass vapor flux in convection