behaviors of solutions for a singular diffusion equation abstract this paper is devoted to some behaviors of solutions of the initial boundary problem for a singular diffusion equation namely localization and large time behavior after given some special explicit solutions it is proved that solutions of the problem possess the localization property next l 2 decay estimate as t is proved by a rather standard energy method finally by comparison with a special solution the expected l decay estimate is derived

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

parabolic equations with nonlinear singularities abstract we show the existence of positive solutions u l 2 0 t h 0 1 for nonlinear parabolic problems with singular lower order terms of the asymptote type more precisely we shall consider both semilinear problems whose model is u t u u 1 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t and quasilinear problems having natural growth with respect to the gradient whose model is u t u u 2 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t with 0 moreover we prove a comparison principle and as an application we study the asymptotic behavior of the solution as t goes to infinity

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

self similar intermediate asymptotics for a degenerate parabolic filtration absorption equation the equation tu u xx2u c 1 xu 2 is known in literature as a qualitative mathematical model of some biological phenomena here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock therefore we refer to this equation as a filtration absorption equation a family of self similar solutions to this equation is constructed numerical investigation of the evolution of non self similar solutions to the cauchy problems having compactly supported initial conditions is performed numerical experiments indicate that the self similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state identical zero an open problem caused by the nonuniqueness of the solution of the cauchy problem is discussed

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

on the one dimensional p laplacian with a singular nonlinearity abstract in this paper we are concerned with the existence and nonexistence of positive solutions of boundary value problems for the one dimensional p laplacian with a singular nonlinearity in the case of the model equation we give the necessary and sufficient conditions of the existence of positive solutions for both the dirichlet problem and the periodic problem

positive solutions of dirichlet and homoclinic type for a class of singular equations abstract we study a nonlinear singular boundary value problem and prove that depending on a relationship between exponents of power terms the problem has either solutions of dirichlet type or homoclinic solutions we make use of shooting techniques and lower and upper solutions

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

existence of positive solutions for the one dimensional singular p laplacian this paper concerns the positive solutions of boundary value problems for the one dimensional singular p laplacian by the classical method of elliptic regularization we obtain some existence results which generalize some results of w zhou x wei positive solutions to bvps for a singular differential equation nonlinear anal 2006 doi 10 1016 j na 2006 06 015

superconvergence for control volume mixed finite element methods on rectangular grids we consider control volume mixed finite element methods for the approximation of second order elliptic problems on rectangular grids these methods associate control volumes covolumes with the vector variable as well as the scalar obtaining local algebraic representation of the vector equation e g darcy s law as well as the scalar equation e g conservation of mass we establish o h 2 superconvergence for both the scalar variable in a discrete l 2 norm and the vector variable in a discrete h rm div norm the analysis exploits a relationship between control volume mixed finite element methods and the lowest order raviart thomas mixed finite element methods

global analysis of piecewise linear systems using impact maps and surface lyapunov functions this paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems pls this methodology infers global properties of pls solely by studying the behavior at switching surfaces associated with pls the main idea is to analyze impact maps i e maps from one switching surface to the next switching surface such maps are known to be unfriendly maps in the sense that they are highly nonlinear multivalued and not continuous we found however that an impact map induced by an linear time invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state this representation of impact maps allows the search for surface lyapunov functions sulf to be done by simply solving a semidefinite program allowing global asymptotic stability robustness and performance of limit cycles and equilibrium points of pls to be efficiently checked this new analysis methodology has been applied to relay feedback on off and saturation systems where it has shown to be very successful in globally analyzing a large number of examples in fact it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology examples analyzed include systems of relative degree larger than one and of high dimension for which no other analysis methodology could be applied this success in globally analyzing certain classes of pls has shown the power of this new methodology and suggests its potential toward the analysis of larger and more complex pls

positive solutions to bvps for a singular differential equation abstract in this paper we are interested in the positive solutions to a singular differential equation 2 f t 0 0 t 1 subject to the first or all of the following boundary conditions 1 0 0 1 0 0 where is a positive constant f t c 1 0 1 f t 0 on 0 1 by the classical method of elliptic regularization some existence results of the above bvps are obtained moreover the uniqueness and the asymptotic behaviors as 0 and as will also be studied

radon measure valued solutions for a class of quasilinear parabolic equations initial value problems for quasilinear parabolic equations having radon measures as initial data have been widely investigated looking for solutions which for positive times take values in some function space in contrast it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the radon measures of the initial data we call such solutions measure valued in contrast to function valued solutionspreviously considered in the literature we first show that there is a natural notion of measure valued solution of problem p below in spite of its nonlinear character a major consequence of our definition is that if the space dimension is greater than one the concentrated part of the solution with respect to the newtonian capacity is constant in time subsequently we prove that there exists exactly one solution of the problem such that the diffuse part with respect to the newtonian capacity of the singular part of the solution with respect to the lebesgue measure is concentrated for almost every positive time on the set where the regular part with respect to the lebesgue measure is large moreover using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time finally the regularity problem is addressed as we give conditions depending on the space dimension the initial data and the rate of convergence at infinity of the nonlinearity to ensure that the measure valued solution of problem p is in fact function valued

mixed covolume methods on rectangular grids for elliptic problems we consider a covolume method for a system of first order pdes resulting from the mixed formulation of the variable coefficient matrix poisson equation with the neumann boundary condition the system may be used to represent the darcy law and the mass conservation law in anisotropic porous media flow the velocity and pressure are approximated by the lowest order raviart thomas space on rectangles the method was introduced by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems reservoir simulation research corporation denver co 1995 as a control volume mixed method and has been extensively tested by jones a mixed finite volume elementary method for accurate computation of fluid velocities in porous media university of colorado at denver 1995 and cai et al computational geosciences 1 1997 pp 289 345 we reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures

positive solutions to a singular differential equation of second order abstract this paper concerns the existence of positive solutions to a singular differential equation u u t u 2 u f t 0 0 t 1 with the boundary conditions u 1 u 0 0 where 0 f t c 0 1 and f t 0 on 0 1 by theories of ordinary differential equations bertsch and ughi m bertsch m ughi positivity properties of viscosity solutions of a degenerate parabolic equation nonlinear anal 14 1990 571 592 proved that when n 1 n is a positive integer and f 1 the problem admits a decreasing positive solution in this paper we show by the classical method of elliptical regularization that if 1 2 1 then the above problem admits at least a positive solution which is not decreasing as a by product of the results the problem with n 1 and f 1 admits at least two positive solutions if n 2

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

nonuniqueness of solutions for a singular diffusion problem in this paper we consider a singular diffusion problem and show by constructing a counterexample that the weak solution to the problem is not unique the proof consists of several steps first we prove that there exists a maximal weak solution to the problem we show that the support of the continuous maximal weak solution cannot decrease in time then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data