tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

boundary regularity on the parabolic monge ampere equation abstract this paper establishes boundary estimates for solutions to the parabolic monge ampere equation u t det d 2 u 1 n f x t

existence results for anisotropic quasilinear parabolic equations with time dependent exponents and gradient term abstract the cauchy dirichlet problem for anisotropic parabolic equation with time dependent exponents and gradient term is considered we state conditions that guarantee the existence and uniqueness of a viscosity solution which is lipschitz continuous in the space variables and holder continuous in time

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

existence of solutions for nonlinear parabolic problem with p x growth abstract in this paper we study the existence of solutions for nonlinear parabolic initial boundary value problem with p x growth conditions in the space w 1 x l p x q l 0 t l 2 and give an existence theorem of weak solutions for the following equation u t a u f where a u div a x t u u a 0 x t u u a x t u u and a 0 x t u u satisfy p x growth conditions with respect to u and u and f w 1 x l q x q

existence and uniqueness of weak solutions for a non uniformly parabolic equation in this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial boundary value problem of a non uniformly parabolic equation some well known parabolic equations are the special cases of this equation

singular and degenerate anisotropic parabolic equations with a nonlinear source abstract the dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered we state conditions that guarantee the existence and uniqueness of a global weak solution to the problem a similar result is proved for the parabolic p laplace equation

the regularity theory for the parabolic double obstacle problem in this paper we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator the result in this paper are generalizations of the theory for the elliptic problem in lee et al calc var partial differ equ 58 3 104 2019 and lee and park the regularity theory for the double obstacle problem for fully nonlinear operator 2018 to parabolic case and also the theory for the parabolic single obstacle problem in caffarelli et al j am math soc 17 4 827 869 2004 to double obstacle case new difficulties in the theory which are generated by the characteristic of parabolic pdes and the existence of the upper obstacle are discussed in detail furthermore the thickness assumptions to have the regularity of the free boundary are carefully considered

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

alternating direction collocation for separable elliptic partial differential equations a complete theory is given for the convergence of alternating direction collocation adc applied to a large class of linear separable elliptic partial differential equations on a rectangular domain it is shown that adc using bicubic hermite splines converges in this setting and converges to the full collocation solution moreover it is shown that a complex embedding of the problem is essential to the theory and that the associated eigensplines and generalized eigensplines can be complex with complex eigenvalues whose geometric and algebraic multiplicities need not be equal nonetheless only real computations are required

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

spline collocation differentiation matrices spline collocation differentiation matrices arising when solving pdes with periodic boundary conditions have block circulant structure in this paper we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices we use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic pdes

ale method for solidification modelling abstract in this paper an arbitrary lagrangian eulerian ale formulation has been developed for the simulation of casting processes the method is applied both to mould filling simulation where it can provide accurate free surface description and to the study of thermo mechanical phenomena occurring in the subsequent cooling down of cast parts prediction of solute transport and of distortions and stresses in the first three sections all governing equations constitutive equations momentum energy solute transport are given then the ale formulation is presented mesh updating with evolving free surfaces using conservative normal vectors to enforce mass conservation treatment of advection terms using an original nodal upwind method definition of lagrangian and eulerian lagrangian regions finally examples of applications are given

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

elliptic equations on convex domains with nonhomogeneous dirichlet boundary conditions abstract in this paper we will study the differentiability on the boundary of solutions of elliptic non divergence differential equations on convex domains the results are divided into two cases i at the boundary points where the blow up of the domain is not the half space if the boundary function is differentiable then the solution is differentiable ii at the boundary points where the blow up of the domain is the half space the differentiability of the solution needs an extra dini condition for the boundary function counterexample is given to show that our results are optimal

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

parabolic equations in time dependent reifenberg domains in this paper we define time dependent parabolic reifenberg domains and study lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains the basic assumption is that the principal coefficients are of parabolic bmo space with small parabolic bmo seminorms it is shown that lp estimates hold for time dependent parabolic d reifenberg domains

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

on a class of fully nonlinear flows in kahler geometry in this paper we study a class of fully nonlinear metric flow on kahler manifolds which includes the j flow as a special case we provide a sufficient and necessary condition for the long time convergence of the flow generalizing the result of songweinkove as a consequence under the given condition we solved the corresponding euler equation which is fully nonlinear of monge ampere type as an application we also discuss a complex monge ampere type equation including terms of mixed degrees which was first posed by chen

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

on the continuity of solutions of the nonhomogeneous evolution p x t laplace equation abstract we study the continuity of solutions of the homogeneous dirichlet problem for the equation u t div u p x t 2 u f in q t 0 t where r n n 2 is a domain with the boundary c 2 and p x t is a given function such that 2 n n 2 p p x t p p c o n s t it is shown that if p f are holder continuous in q t and u 0 p x 0 l 1 then u c 1 1 2 q t t for every 0 moreover if f is lipschitz continuous in t p p x p l and either p x p 2 or 2 n n 2 p p x p 2 then the solutions are lipschitz continuous in time in q t t

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

finite difference preconditioning for solving orthogonal collocation equations for boundary value problems a technique to construct a low order finite difference preconditioner for solving orthogonal collocation equations for boundary value problems is presented it is shown numerically and theoretically that the spectral condition numbers of the preconditioned collocation matrices are bounded by constants independent of the number of mesh nodes when certain exact low order finite difference preconditionings are used preconditioners based on incomplete lu factorization are also discussed numerical experiments show the efficiency and robustness of the preconditioning

free boundary regularity close to initial state for parabolic obstacle problem here partial derivative p denotes the parabolic boundary h is a parabolic operator with certain properties q is the upper half of the unit cylinder in rn 1 and the equation is satisfied in t

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

a fixed mesh eulerian lagrangian approach for stress analysis in continuous casting we present a method for the analysis of the strains and stresses developed in the solidified portion of a metal strand during the continuous casting process steady state conditions are assumed the solidified metal is modelled as a standard inelastic solid with isotropic hardening and von mises yield criterion constitutive equations are formulated over the material particles lagrangian approach that instantaneously occupy a fixed domain attached to the casting machine which is discretized eulerian approach particle tracking needed to record the deformation history at each fixed sampling point becomes trivial because of the usual hypothesis of constant and homogeneous advection velocity backward euler implicit finite differences and mixed finite elements are used for time and spatial discretization respectively the resulting non linear algebraic equations are solved using the newton raphson method copyright 2006 john wiley sons ltd

worn stones with flat sides all time regularity of the interface we study the all time regularity of the free boundary problem associated to the deformation of a compact weakly convex surface in 3 with a flat side by its gaussian curvature we show that under certain necessary regularity and non degeneracy initial conditions the interface separating the flat from the strictly convex side remains smooth on 0 t tc up to the vanishing time tc of the flat side

alternating direction collocation for rectangular regions abstract an alternating direction procedure for finite element collocation equations can be formulated on rectangular regions finite element collocation offers a great savings over finite element techniques in the matrix formation stage of the numerical process moreover with an alternating direction procedure multidimensional problems can be solved as a series of one dimensional problems this greatly reduces the amount of computer time and storage required to solve large problems

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

fourier analysis of iterative methods for elliptic boundary value problems his paper presents a fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems as in the von neumann stability analysis of hyperbolic and parabolic problems the approach is easier to apply reveals more details about convergence properties than about standard techniques and can be applied in a systematic way to a wide class of numerical methods although the analysis is applicable only to periodic problems the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions such as the dirichlet problem in addition they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the dirichlet case

global higher regularity of solutions to singular p x t parabolic equations abstract we study the homogeneous dirichlet problem for the equation u t div u p x t 2 u f x t u in the cylinder q t 0 t r d d 2 it is assumed that p x t 2 d d 2 2 and p p t are bounded a e in q t we find conditions on p x t f x t u and u x 0 sufficient for the existence of strong solutions local or global in time it is proven that the strong solutions possess the property of global higher regularity u t l 2 q t u l 0 t l 2 d i j 2 u p x t l 1 q t

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

twice differentiability of solutions to fully nonlinear parabolic equations near the boundary abstract in this paper we prove h 2 regularity for viscosity solutions to non convex fully nonlinear parabolic equations near the boundary this constitutes the parabolic counterpart of a similar c 2 regularity result due to silvestre and sirakov proved in 17 for solutions to non convex fully nonlinear elliptic equations

the problem of dirichlet for anisotropic quasilinear degenerate elliptic equations we consider the dirichlet problem for a class of anisotropic degenerate elliptic equations new a priori estimates for solutions and for the gradient of solutions are established based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated the results are new even in the semilinear case when the principal part is the laplace operator

finite element modelling of surface tension effects using a lagrangian eulerian kinematic description abstract an algorithm has been devised to model surface tension forces during fluid motion it makes use of an eulerian formulation applied to the governing equations and a lagrangian treatment of the free surface it is assumed that the effect of air on the free surface movement is negligible thus only the fluid domain is considered during the analysis the free surface is subjected to a surface tension force proportional to the local curvature this manifests itself as a simple and accurate method of modelling surface tension effects by virtue of the free surface being explicitly determined the method is validated by modelling the effect of surface tension on a static rod a further numerical example is used to illustrate the effect of surface tension on a non equilibrium square rod wall adhesion modelling is demonstrated by modelling the wetting of a capillary tube finally the method is used in an industrial context to model the filling of mould cavities for which surface tension forces are important

existence of lipschitz continuous solutions to the cauchy dirichlet problem for anisotropic parabolic equations abstract the cauchy dirichlet and the cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered we show that the time derivative u t of a solution u belongs to l under a suitable assumption on the smoothness of the initial data moreover if the domain satisfies some additional geometric restrictions then the spatial derivatives u x i belong to l as well in the singular case we show that the second derivatives u x i x j of a solution of the cauchy problem belong to l 2

lateral boundary differentiability of solutions of parabolic equations in nondivergence form abstract the lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior dini condition and is punctually c 1 differentiable one sided in t direction the classical barrier technique the maximum principle the interior harnack inequality and an iteration procedure are the main analytical tools

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

differentiability at lateral boundary for fully nonlinear parabolic equations abstract for fully nonlinear uniformly parabolic equations the first derivatives regularity of viscosity solutions at lateral boundary is studied under new dini type conditions for the boundary which is called reifenberg dini conditions and is weaker than usual dini conditions

a global approach to fully nonlinear parabolic problems we consider the general initial boundary value problem 1 u t f x t u du du f x t x t qt 0 t 2 g x t u du g x t x t st 0 t 3 u x 0 h x x where is a bounded open set in rn with sufficiently smooth boundary the problem 1 3 is first reduced to the analogous problem in the space w 4 0 p qt with zero initial condition and f w 2 0 p qt g w 3 1 p 0 p st the resulting problem is then reduced to the problem au 0 where the operator a w 4 0 p qt w 4 0 p qt satisfies condition s this reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in sobolev spaces the local and global solvability of the operator equation au 0 are achieved via topological methods developed by i v skrypnik further applications are also given involving relevant coercive problems as well as galerkin approximations

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100