convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

maximum principle satisfying second order discontinuous galerkin schemes for convection diffusion equations on triangular meshes we propose second order accurate discontinuous galerkin dg schemes which satisfy a strict maximum principle for general nonlinear convection diffusion equations on unstructured triangular meshes motivated by genuinely high order maximum principle satisfying dg schemes for hyperbolic conservation laws perthame 1996 and zhang 2010 14 26 we prove that under suitable time step restriction for forward euler time stepping for general nonlinear convection diffusion equations the same scaling limiter coupled with second order dg methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy similar to the purely convection cases the limiters are mass conservative and easy to implement strong stability preserving ssp high order time discretizations will keep the maximum principle following the idea in zhang 2012 30 we extend the schemes to two dimensional convection diffusion equations on triangular meshes there are no geometric constraints on the mesh such as angle acuteness numerical results including incompressible navier stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods

iterative identification of block oriented nonlinear systems based on biconvex optimization abstract we investigate the identification of a class of block oriented nonlinear systems which is represented by a common model in this paper then identifying the common model is formulated as a biconvex optimization problem based on this a normalized alterative convex search nacs algorithm is proposed under a given arbitrary nonzero initial condition it is shown that we only need to find the unique partial optimum point of a biconvex cost function in order to obtain its global minimum point thus the convergence property of the proposed algorithm is established under arbitrary nonzero initial conditions by applying the results to hammerstein wiener systems with an invertible nonlinear function the long standing problem on the convergence of iteratively identifying such systems under arbitrary nonzero initial conditions is also now solved

a geometric discretization and a simple implementation for variational mesh generation and adaptation we present a simple direct discretization for functionals used in the variational mesh generation and adaptation meshing functionals are discretized on simplicial meshes and the jacobian matrix of the continuous coordinate transformation is approximated by the jacobian matrices of affine mappings between elements the advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional moreover the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form which allows a simple implementation of variational mesh generation and adaptation on computer since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods the result in this work can be used to develop simple implementations of those methods numerical examples are given

a priori l r error estimates for galerkin approximations to porous medium and fast diffusion equations galerkin approximations to solutions of a cauchy dirichlet problem governed by the generalized porous medium equation formulamath on bounded convex domains are considered the range of the parameter includes the fast diffusion case 1 2 using an euler finite difference approximation in time the semi discrete solution is shown to converge to the exact solution in l 0 t l norm with an error controlled by o t1 4 for 1 2 and o t1 2 for 2 for the fully discrete problem a global convergence rate of o t1 4 in l 2 0 t l norm is shown for the range 2n n 1 2 for 2 a rate of o t1 2 is shown in l 0 t l norm

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

study of the solutions to a model porous medium equation with variable exponent of nonlinearity abstract the authors of this paper study the dirichlet problem of the following equation u t div u x t u f u p x t 1 u the existence and uniqueness of weak solutions are proven also the properties of the solutions are studied which include the property of finite speed of propagation of disturbances localization property and the property of vanishing at a finite time etc

convergence of the finite element method for the porous media equation with variable exponent in this work we study the convergence of the finite element method when applied to the following parabolic equation u t div u gamma mathbf x nabla u f mathbf x t mathbf x in omega subset r m t in 0 t since the problem may be of degenerate type we utilize an approximate problem regularized by introducing a parameter varepsilon we prove under certain conditions on gamma and f that the weak solution of the approximate problem converges to the weak solution of the initial problem when the parameter varepsilon tends to zero discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved finally we present some numerical results of a matlab implementation of the method

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

numerical methods for flows through porous media i the degenerate parabolic equation aut v oulvvu v 1 has been used to model the flow of gas through a porous medium error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved and several new regularity results are given 1 a porous medium equation introduction we shall study the porous medium equation 1 1 au at v i u i vu on 2 x 0 t 1 2 au an 0 on aa x o t 1 3 u x 0 uo x on 2 where v 1 is a parameter and q is a bounded domain in rn n 3 with a smooth boundary the initial function u0 is assumed to be nonnegative and four times continuously differentiable on u notice that the compatibility condition auo an 0 holds on asa our main result is the derivation of error estimates for numerical approximations to the problem 1 1 1 3 which we shall refer to as the porous medium equation or ipme the pme does not in general admit classical solutions existence and uniqueness of weak solutions was proved in one space dimension by oleinik kalashnikov and czou 15 16 and in several space dimensions by lions 12 these proofs concern the pme with different boundary conditions but the arguments carry over to the pme 1 1 1 3 the maximum principle implies that since u0 is nonnegative on 92 u x t is nonnegative for all x t e q x 0 t see 15 16 if u0 is nonzero the neumann boundary condition implies that u will eventually become strictly positive and 1 1 will become nondegenerate for all time t to to sufficiently large we can rewrite 1 1 in the form 1 4 au at ak u on qx 0 t received may 11 1979 revised july 24 1981 1980 mathematics subject classification primary 35k65 65n30 65n15 current address reservoir engineering branch petroleum engineering division bp exploration dos britannic house moore lane london ec2y 9bu united kingdom d1983 american mathematical society 0025 5718 82 0000 1 063 08 25 435 this content downloaded from 157 55 39 55 on tue 23 aug 2016 04 12 54 utc all use subject to http about jstor org terms

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

numerical study of the porous medium equation with absorption variable exponents of nonlinearity and free boundary abstract in this paper we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2d domains with moving boundaries the boundary s movement is governed by an equation prompted by the darcy law and the spatial discretization is defined by a triangulation of the domain at each finite element the solution is approximated by piecewise polynomial functions of degree r 1 using lagrange interpolating polynomials in area coordinates the vertices of the triangles move according to a system of differential equations which is added to the equations of the problem the resulting system is converted into a system of ordinary differential equations in time variable which is solved using a suitable integrator the integrals that arise in the system of ordinary differential equations are calculated using the gaussian quadrature finally we present some numerical results of application of this technique

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

statistical mechanics perspective on the phase transition in vertex covering offinite connectivity random graphs the vertex cover problem is studied for random graphs gn cn having n vertices and cn edges exact numerical results are obtained by a branch and bound algorithm it is found that a transition in the coverability at a c dependent threshold x xc c appears where xn is the cardinality of the vertex cover this transition coincides with a sharp peak of the typical numerical e1ort which is needed to decide whether there exists a cover with xn vertices or not for small edge concentrations c two less than arrows one inside the other 0 5 a cluster expansion is performed giving very accurate results in this regime these results are extended using methods developed in statistical physics the so called annealed approximation reproduces a rigorous bound on xc c which was known previously the main part of the paper contains an application of the replica method within the replicasymmetric ansatz the threshold xc c and various statistical properties of minimal vertex covers can be calculated for c e 2 the results show an excellent agreement with the numerical findings at average vertex degree 2c e an instability of the simple replica symmetric solution occurs

finite element approximation of the fast diffusion and the porous medium equations 1the fast diffusion equation u t delta u m 1 u 0 m 1 and the porous medium equation 1 m infty are studied in a parabolic cylinder omega times 0 t a fully discrete galerkin approximation is considered using c 0 piecewise linear finite elements in space and the backward euler time discretization a priori error estimates in quasi norms and rates of convergence are proved

interfaces in multidimensional diffusion equations with absorption terms we study the properties of interfaces in solutions of the cauchy problem for the nonlinear degenerate parabolic equation ut um aup in rn 0 t n 1 2 3 with the parameters m 1 p 0 a 0 satisfying the condition m p 2 we show that the velocity of the interface t supp u x t is given by the formula v m m 1 um 1 t where is the solution of the degenerate elliptic equation div u aup 0 0 on t the first term expresses the classical darcy law while the second one models the presence of an external force that makes the interface to move in the inward direction we give explicit formulas which represent the interface t as a bijection from 0 it is proved that the solution u and its interface t are analytic functions of time t and that they preserve the initial regularity in the spatial variables we also show that the regularity of the interface velocity v with respect to the spatial variables is better than it was at the initial instant the analysis is based on a special coordinate transformation a local version of lagrangian coordinates that renders the free boundary stationary

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

error estimates for a class of degenerate parabolic equations the degenerate parabolic equation u t delta beta u is considered in the cylinder omega times 0 t where omega subset r n is bounded beta 0 beta 0 0 and beta geq0 e g beta u u u m 1 m 1 a fully discrete scheme is given using c0 piecewise linear finite elements in space and a backward difference scheme in time l2 error estimates for the approximation of u and of beta u are proven for that purpose we employ the fact that the free boundary has a finite speed of propagation

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach