an adaptive grid finite difference method for conservation laws abstract adaptive grid finite difference methods for computing time accurate solutions of nonlinear hyperbolic conservation laws in one space dimension are studied the basic approach is to decouple the determination of the moving grid from the solution of the differential equation on the moving grid the grid is determined by an elliptic grid generation technique suitably modified for time accurate computations two methods godunov s scheme and an artificial viscosity method are used to discretize the differential equation the technique is applied to problems involving the buckley leverett equation modelling the flow of two immiscible fluids in a porous medium substantial improvement in computational efficiency is obtained by using the adaptive grid technique extensions of this approach to several space dimensions and systems of equations are discussed

accuracy analysis of an adaptive mesh refinement method using benchmarks of 2 d steady incompressible lid driven cavity flows and coarser meshes lid driven cavity flows have been widely investigated and accurate results have been achieved as benchmarks for testing the accuracy of computational methods this paper verifies the accuracy of an adaptive mesh refinement method numerically using 2 d steady incompressible lid driven cavity flows and coarser meshes the accuracy is shown by verifying that the centres of vortices given in the benchmarks are located in the refined grids of refined meshes for reynolds numbers 100 1000 and 2500 using coarser meshes the adaptive mesh refinement method performs mesh refinement based on the numerical solutions of navier stokes equations solved by a finite volume method with a well known simple algorithm for pressure velocity coupling the accuracy of the refined meshes is shown by comparing the profiles of horizontal and vertical components of velocity fields with the corresponding components of the benchmarks together and drawing closed streamlines the adaptive mesh refinement method verified in this paper can be applied to find the accurate numerical solutions of any mathematical models containing continuity equations for incompressible fluid steady state fluid flows or mass and heat transfer

excursions for polymers in elongational flows the study of the configurational statistical mechanics of polymers in elongational flows requires the evaluation of quadratic path integrals here we present a new technique for the evaluation of such path integrals the method relies on ito excursion theory and the ray knight theorem from probability theory in addition to providing a powerful computational method in standard cases the method generalizes to deal with quadratic path integrals for branching processes

a mixed variable continuously deforming finite element method for parabolic evolution problems part i the variational formulation for a single evolution equation the objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow fluid flow and species transport processes with sharply defined phase change interfaces in this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non linear time dependent partial differential equations in part ii of this paper1 we extend the formulation given here for a single evolution equation to a system of coupled evolution equations in part iii2 we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method

a higher order tv type variational problem related to the denoising and inpainting of images abstract we give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging the overall aim of this note is to investigate if and in which manner results from the first author s previous work on variants of the tv regularization model see e g bildhauer and fuchs 2012 2014 2015 and fuchs and tietz 2015 can be extended to functionals which involve higher derivatives this seems to be not only of theoretical interest but also relevant to applications since higher order tv denoising appears to maintain the advantages of the classical model as introduced by rudin osher and fatemi in rudin et al 1992 while avoiding the unpleasant staircasing effect see e g bredies et al 2010 or lysaker et al 2003 our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation dual solutions as well as partial regularity of minimizers

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

a study of moving mesh methods applied to a thin flame propagating in a detonator delay element we study the application of moving mesh methods to a one dimensional time dependent detonator delay element problem we consider moving mesh methods based on the equidistribution principle derived by huang et al 1 adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients significant improvements in accuracy and efficiency are achieved by adapting the nodes mesh points so that they are concentrated about areas of large solution variations each system of equations for the moving mesh methods is solved in conjunction with the detonator problem in this paper the system of ordinary differential equations that results after discretising in space is solved using the double precision version of the stiff ordinary differential equation solver dassl the numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method

semiflexible polymers in straining flows we introduce a model for semiflexible polymer chains based on the integral of an appropriate gaussian process the stiffness is characterized physically by adding a bending energy the degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity the limiting case reduces to the brownian model of completely flexible chains studied in earlier work the calculation of the partition function for the configuration statistical mechanics i e the distribution of shapes of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated gaussian process an exact formula for the partition function is presented however in practice this formula is too complicated for most computations we therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first order term which gives the first order deviation from the completely flexible case in addition to the partition function the method presented here can also deal with other quadratic functionals such as the stochastic area associated with two polymer chains

a particle method for collisional kinetic equations i basic theory and one dimensional results abstract a new method is introduced for describing the collisional term in the framework of a particle scheme the equations of motion of the particles contain an additional term which takes into account the collisions numerical techniques are shown to compute this extra term and to solve the equations exact results concerning stability and consistency of the schemes for the diffusion equation are derived some numerical results for the one dimensional case are reported

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

lattice boltzmann model for the modified burgers equation abstract in this paper a special lattice boltzmann model is developed to solve the modified burgers equation the maximum principle and the stability are proved the model has been verified by several test examples excellent agreement is obtained between numerical predictions and the constructed asymptotic solutions or available numerical results the careful numerical study indicates the computational efficiency and stability of the lattice boltzmann method

a finite difference moving mesh method based on conservation for moving boundary problems we propose a velocity based moving mesh method in which we move the nodes so as to preserve local mass fractions consequently the mesh evolves to be finer where the solution presents rapid changes naturally providing higher accuracy without the need to add nodes we use an integral approach which avoids altering the structure of the original equations when incorporating the velocity and allows the solution to be recovered algebraically we apply our method to a range of one dimensional moving boundary problems the porous medium equation richards equation and the crank gupta problem we compare our results to exact solutions where possible or to results obtained from other methods and find that our approach can be very accurate 1 relative error with as few as ten or twenty nodes

mesh independent spectra in the moving finite element equations we derive the moving finite element mfe equations for the solution of a scalar evolutionary equation in d space dimensions d geqq 1 and introduce the elementwise approach to mfe this approa

improved error estimation of dynamic finite element methods for second order parabolic equations dynamic finite element schemes are analyzed for second order parabolic problems these schemes permit different finite element spaces at different time levels in order to efficiently capture time changing localized phenomena such as moving sharp fronts or layers the dynamical change of grids and interpolation polynomials is necessary and essential to many large scale transient problems standard characteristic and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems in a unified framework with improved a priori error estimates and convergence results

a geometrical mechanical interpretation of gradient weighted moving finite elements the usual explanation of the gradient weighted moving finite element gwmfe method has been in terms of its variational interpretation this paper presents a more intuitive geometrical mechanical interpretation of gwmfe as a balance of forces on the nodes forces concentrated onto the nodes by the laws of leverage it also presents significant simplifications in the internodal viscosity terms for regularization of the nodal movements plus some simple linear internodal tensions for regularization of the long term nodal positioning these simplifications of the regularizations are especially important in two and three space dimensions one of the generalizations which follows from the geometrical mechanical interpretation is a promising but still untested second gwmfe formulation for systems of pdes the original mfe method is seen to be the small slope limit of gwmfe under vertical rescaling reporting on the design and extensive numerical trials of robust and versatile gwmfe systems codes in one and two dimensions is deferred to two forthcoming papers by carlson and the author design and application of a gradient weighted moving finite element code part i in 1 d siam j sci comput to appear and design and application of a gradient weighted moving finite element code part ii in 2 d siam j sci comput to appear here only a few illustrative examples are presented involving motion of surfaces by mean curvature i e by surface tension

two dimensional fully adaptive solutions of solid solid alloying reactions abstract solid solid alloying reactions occur in a variety of pyrotechnical applications they arise when a mixture of powders composed of appropriate oxidizing and reducing agents is heated the large quantity of heat evolved produces a self propagating reaction front that is often very narrow with sharp changes in both the temperature and the concentrations of the reacting species solution of problems of this type with an equispaced or mildly nonuniform grid can be extremely inefficient in this paper we develop a two dimensional fully adaptive method for solving problems of this class the method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval in each direction at each time level we monitor the solution from one time level to another to ensure that the local error per unit step associated with the time differencing method is below some specified tolerance the method is applied to several examples involving exothermic diffusion controlled self propagating reactions in packed bed reactors

a self adaptive moving mesh method for the camassa holm equation a self adaptive moving mesh method is proposed for the numerical simulations of the camassa holm equation it is an integrable scheme in the sense that it possesses the exact n soliton solution it is named a self adaptive moving mesh method because the non uniform mesh is driven and adapted automatically by the solution once the non uniform mesh is evolved the solution is determined by solving a tridiagonal linear system due to these two superior features of the method several test problems give very satisfactory results even if by using a small number of grid points

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

regularity results for equilibria in a variational model for fracture in recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms of the form f vu dx i n x jn k where q is an open bounded subset of r q 1 g l r a 0 0 the bulk energy density f is quasiconvex k c r is closed and the admissible deformation u q f r is c in i k one of the main issues has to do with regularity properties of the crack site k for a minimizing pair k u in the scalar case i e when u o f r similar models were adopted to image segmentation problems and the regularity of the edge set k has been successfully resolved for a quite broad class of convex functions f with growth p 1 at infinity in turn this regularity entails the existence of classical solutions the methods thus used cannot be carried out to the vectorial case except for a very restrictive class of integrands in this paper we deal with a vector valued case on the plane obtaining regularity for minimizers of q corresponding to polyconvex bulk energy densities of the form where the convex function h grows linearly at infinity 1991 mathematics subject classification amer math soc 35j20 49q20 49j45 49n60

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

exponential splitting time integration for pseudospectral methods on moving meshes moving meshes are successfully used in many fields here we investigate how a recently proposed approach to combine the strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes a double representation of a function by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula is an essential part of the numerical integration before numerical implementation the original pde is transformed into a suitable form the approach is illustrated on the linear heat transfer equation

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

solution of parabolic equations by backward euler mixed finite element methods on a dynamically changing mesh we develop and analyze methods based on combining the lowest order mixed finite element method with backward euler time discretization for the solution of diffusion problems on dynamically changing meshes the methods developed are shown to preserve the optimal rate error estimates that are well known for static meshes the novel aspect of the scheme is the construction of a linear approximation to the solution which is used in projecting the solution from one mesh to another extensions to advection diffusion equations are discussed where the advection is handled by upwinding numerical results validating the theory are also presented

moving finite element model for one dimensional infiltration in unsaturated soil the numerical integration of richards equation for infiltration into homogeneous unsaturated soil is carried out by a moving finite element mfe method accurate solution of this one dimensional nonlinear partial differential equation by standard finite difference fd or finite element fe methods can be obtained by using a fine discretization on space and time this can lead to large computer times with the mfe method grid points are moved during computation along the wetting front so that accuracy can be maintained by using a small number of nodes for simulating cases several meters in depth for these cases the mfe method is faster than fd and fe methods the numerical mfe model is verified through comparison with the quasi analytical solution by philip 1957 moreover we compare for some other test problems the computational efficiency of this method with that of two efficient pressure head based fd and fe fixed grid formulations

applications of the finite element method to modeling the atmospheric boundary layer abstract the finite element method is used to describe the detailed properties of the atmospheric boundary layer by use of a high resolution model and its bulk properties by use of a simple vertically integrated model features of the finite element method that can be exploited for applications to the atmospheric boundary layer include the capability to use different basis functions in different parts of the domain ability to grid over irregular terrain ease of using time dependent basis functions and the natural way that surface boundary conditions and vertically integrated properties enter the model

time stepping schemes for moving grid finite elements applied to reaction diffusion systems on fixed and growing domains in this paper we illustrate the application of time stepping schemes to reaction diffusion systems on fixed and continuously growing domains by use of finite element and moving grid finite element methods we present two schemes for our studies namely a first order backward euler finite differentiation formula coupled with a special form of linearisation of the nonlinear reaction terms 1 sbem and a second order semi implicit backward finite differentiation formula 2 sbdf with no linearisation of the reaction terms our results conclude that for the type of reaction diffusion systems considered in this paper the 1 sbem is more stable than the 2 sbdf scheme and that the 1 sbem scheme has a larger region of stability at least by a factor of 10 than that of the 2 sbdf scheme as a result the 1 sbem scheme becomes a natural choice when solving reaction diffusion problems on continuously deforming domains

an adaptive moving grid method for one dimensional systems of partial differential equations we describe a fully adaptive moving grid method for solving initial boundary value problems for systems of one space dimensional partial differential equations whose solutions exhibit rapid variations in space and time the method based on finite differences is of the lagrangian type and has been derived through a co ordinate transformation which leads to equidistribution in space of the second derivative our technique is intermediate between static regridding methods where nodes remain fixed for intervals of time and continuously moving grid methods where the node movement and the pde integration are fully coupled in our approach the computation of the moving grids and the solution on these grids are carried out separately while the nodes are moved at each time step two error monitors have been implemented one to govern the time step selection and the other to eventually adapt the number of moving nodes the method allows the use of different moving grids for different components in the pde system numerical experiments are presented for a set of five sample problems from the literature including two problems from combustion

an error indicator monitor function for an r adaptive finite element method an r adaptive finite element method based on moving mesh partial differential equations pdes and an error indicator is presented the error indicator is obtained by applying a technique developed by bank and weiser to elliptic equations which result in this case from temporal discretization of the underlying physical pdes on moving meshes the construction of the monitor function based on the error indicator is discussed numerical results obtained with the current method and the commonly used method based on solution gradients are presented and analyzed for several examples c 2001 academic press

mixed moving finite element method abstract in this paper we present a mixed moving finite element method mmfem in this method nodes are divided into several classes and move differently we concentrate on burgers equation with a linear basis function and carry out the element and global analysis the computation shows the improvement in contrast to moving finite element method

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

a mixed variable continuously deforming finite element method for parabolic evolution problems part ii the coupled problem of phase change in porous media the conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed variable finite element computational scheme for a single evolution equation was presented in part i 1 in this paper part ii we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two dimensional transport problems governed by a system of coupled non linear partial differential evolution equations in particular we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts and abrupt discontinuous moving phase change interfaces in this method the potentials such as the temperature pressure and species concentration and the corresponding fluxes are permitted to jump in value across the phase change interfaces the equations and the jump conditions governing the physical phenomena which were specialized from a general multiphase multiconstituent mixture theory provided the basis for the development and implementation of a two dimensional numerical simulator this simulator can effectively resolve steep continuous fronts i e shock capturing without oscillations or numerical dispersion and can accurately represent and track discontinuous fronts i e shock fitting through adaptive grid deformation and redistribution the numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in part iii2 of this paper

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

a two dimensional mesh moving technique for time dependent partial differential equations abstract we discuss an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial boundary value problems for systems of partial differential equations in two space dimensions and time the mesh moving technique is based on an algebraic node movement function determined from the geometry and propagation of regions having significant discretization error indicators our procedure is designed to be flexible so that it can be used with many existing finite difference and finite element methods to test the mesh moving algorithm we implemented it in a system code with an initial mesh generator and a maccormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws results are presented for several computational examples the moving mesch scheme reduces dispersive errors near shocks and wave fronts and thereby reduces grid requirements necessary to compute accurate solutions while increasing computational efficiency

on the mass matrix spectrum bounds of wathen and the local moving finite elements of baines andrew wathen has shown that the eigenvalues of the diagonally preconditioned piecewise linear moving finite element mfe or finite element fe mass matrix in n dimensions lie in frac 1 2 1 frac 1 2 n baines using similar considerations has designed very local mfe methods with block diagonal mass matrices in this paper a simplified proof of wathen s basic spectrum bound is given it results from a simple comparison bound between the l 2 norm and a certain diagonal norm for discontinuous piecewise linear functions on an arbitrary triangular grid the simplicity of our comparison lets us extend wathen s result to other situations such as miller s gradient weighted mfe method gwmfe it also lets us design very local mfe and fe and gwmfe methods which minimize the pde residual dot u l u not in l 2 norm but in a comparable norm these methods turn out to be equivalent to those of baines these methods retain the desired conservation properties for pde s in conservation l

a constrained moving finite element solution of the one dimensional oxygen diffusion with absorption problem abstract we derive an adaptive finite element method for the numerical solution of a one dimensional implicit moving boundary problem the oxygen diffusion with absorption problem of crank and gupta the method which includes a moving grid and incorporates an iteration for the velocity of the moving boundary produces good results over a range of parameters we present numerical and graphical results in one dimension and indicate an extension of the method to two dimensions

implicit method for the computation of unsteady flows on unstructured grids an implicit method for the computation of unsteady flows on unstructured grids is presented following a finite difference approximation for the time derivative the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure the method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage inviscid and viscous unsteady flows are computed to validate the procedure the issue of the mass matrix which arises with vertex centered finite volume schemes is addressed the present formulation allows the mass matrix to be inverted indirectly a mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies as an example of flow over bodies in relative motion flow over a multi element airfoil system undergoing deployment is computed

a moving mesh finite element method for the two dimensional stefan problems an r adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two dimensional heat conduction problems with a phase change the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material the enthalpy equation is discretised using a semiimplicit galerkin finite element method using linear basis functions the moving finite element method is applied to problems where the phase front is cusp shaped and where the interface changes topology

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

a variable node finite element method abstract a new finite element method with variable nodes for integrating partial differential equations in one space dimension is presented the method utilizes the approximation power of piecewise polynomial functions better than a fixed node finite element method because it distributes grid points in a nonuniform way to suit the peculiarities of the solution at each instant of time the method uses the gradient and curvature of the approximating piecewise polynomial itself to determine the position of the nodes rather than physical properties which vary from problem to problem the method has been implemented in a program vfe which has the exceptional feature of a variable space step as well as a variable time step since the method concentrates nodes in regions where small inter nodal spacing is needed it is very efficient in integrating equations with shock like solutions finally the method can be extended to systems of partial differential equations in one dimension and to equations in two dimensions

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

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

finite element methods for nonlinear flows in porous media abstract a model problem is presented for fluid flow in porous media which has numerical difficulties common with many other areas of nonlinear mechanics special finite element techniques are described which have proved useful in alleviating many of these difficulties a modified method of characteristics mixed finite element methods for fluid velocities singularity removal and self adaptive local grid refinement techniques are presented both theoretical and computational results are described together with an indication of the present and future uses of finite element methods in large scale reservoir simulation

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

an adaptive moving finite volume scheme for modeling flood inundation over dry and complex topography 1 a new geometrical conservative interpolation on unstructured meshes is developed for preserving still water equilibrium and positivity of water depth at each iteration of mesh movement leading to an adaptive moving finite volume amfv scheme for modeling flood inundation over dry and complex topography unlike traditional schemes involving position fixed meshes the iteration process of the afmv scheme moves a fewer number of the meshes adaptively in response to flow variables calculated in prior solutions and then simulates their posterior values on the new meshes at each time step of the simulation the amfv scheme consists of three parts an adaptive mesh movement to shift the vertices position a geometrical conservative interpolation to remap the flow variables by summing the total mass over old meshes to avoid the generation of spurious waves and a partial differential equations pdes discretization to update the flow variables for a new time step five different test cases are presented to verify the computational advantages of the proposed scheme over nonadaptive methods the results reveal three attractive features i the amfv scheme could preserve still water equilibrium and positivity of water depth within both mesh movement and pde discretization steps ii it improved the shock capturing capability for handling topographic source terms and wet dry interfaces by moving triangular meshes to approximate the spatial distribution of time variant flood processes iii it was able to solve the shallow water equations with a relatively higher accuracy and spatial resolution with a lower computational cost

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

a simple moving mesh method for one and two dimensional phase field equations a simple moving mesh method is proposed for solving phase field equations the numerical strategy is based on the approach proposed in li et al j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution the phase field equations are discretized by a finite volume method and the mesh moving part is realized by solving the conventional euler lagrange equations with the standard gradient based monitors numerical results demonstrate the accuracy and effectiveness of the proposed algorithm

optimal discretizations in adaptive finite element electromagnetics one of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently therefore the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies ultimately the analysis of optimally discretized systems in order to learn about ideal solution characteristics can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably a theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one two and three dimensional systems the formulation is based on variational aspects of optimal discretizations for helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems in addition the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time harmonic phenomena moreover finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed so that the implications of the theory for practical h p hp and r type finite element adaption in multidimensional analyses may be examined copyright 2001 john wiley sons ltd

accuracy verification of a 2d adaptive mesh refinement method for incompressible or steady flow locating accurate centres of vortices is one of the accuracy measures for computational methods in fluid flow and the lid driven cavity flows are widely used as benchmarks this paper analyses the accuracy of an adaptive mesh refinement method using 2d steady incompressible lid driven cavity flows the adaptive mesh refinement method performs mesh refinement based on the numerical solutions of navier stokes equations solved by navier2d a vertex centred finite volume code that uses the median dual mesh to form the control volumes cvs about each vertex the accuracy of the refined meshes is demonstrated by the centres of vortices obtained in the benchmarks being contained in the twice refined cells the adaptive mesh refinement method investigated in this paper is proposed based on the qualitative theory of differential equations theoretically infinite refinements can be performed on an initial mesh practically we can stop the process of refinement based on tolerance conditions the method can be applied to find accurate numerical solutions of any mathematical models containing the continuity equation for incompressible fluid or steady state fluid flow

an analysis of the moving finite element procedure the moving finite element procedure of miller siam j numer anal 18 1981 pp 1019 1057 is analysed and broken down into a number of steps the central step is a mapping from element velocities to nodal velocities which is exact for a class of one dimensional scalar nonlinear partial differential equation pdes for more general pdes the mapping is preceded by an element by element projection step in higher dimensions the mapping is followed by node by node projection for first order pdes the procedure gives rise to ordinary differential equation systems approximating the method of characteristics for second order operators the behaviour of the nodes can only be described qualitatively the central mapping is a legendre transformation from the data to the nodal velocities while the full procedure is an approximate legendre transformation the same transformation can be used to construct a dual pde which may be numerically easier to solve the structure is used to analyse nodal movement and to giv

a new adaptive mesh refinement strategy for numerically solving evolutionary pde s a graph based implementation of quadtree meshes for dealing with adaptive mesh refinement amr in the numerical solution of evolutionary partial differential equations is discussed using finite volume methods the technique displays a plug in feature that allows replacement of a group of cells in any region of interest for another one with arbitrary refinement and with only local changes occurring in the data structure the data structure is also specially designed to minimize the number of operations needed in the amr implementation of the new scheme allows flexibility in the levels of refinement of adjacent regions moreover storage requirements and computational cost compare competitively with mesh refinement schemes based on hierarchical trees low storage is achieved for only the children nodes are stored when a refinement takes place these nodes become part of a graph structure thus motivating the denomination autonomous leaves graph alg for the new scheme neighbors can then be reached without accessing their parent nodes additionally linear system solvers based on the minimization of functionals can be easily employed alg was not conceived with any particular problem or geometry in mind and can thus be applied to the study of several phenomena some test problems are used to illustrate the effectiveness of the technique

adaptive mesh techniques for fronts in star formation abstract we present an implicit adaptive grid finite difference technique specifically designed to locate and track arbitrary fronts interfaces and structures in a radiation hydro flow the adaptive mesh is constructed in such a way that the average change of the flow variables per grid zone is the same throughout the entire grid a local grid refinement of up to a factor of 10 6 compared to an equidistant mesh has been obtained on a machine with 14 decimal digits per word crucial to this type of approach is the use of a stiffness operator which prevents tangling of the mesh the elliptic equation for the grid motion is solved implicitly coupled with the equations for the flow variables per timestep very steep and narrow flow features can travel many times their own width in true space however with respect to the adaptive mesh the motion of such flow features is always less than one zone thus the courant friedrichs lewy 1 cfl condition s accuracy limitation is always fulfilled although its stability criterion does not pose any timestep limitations for our implicit numerical method in fact the choice of timesteps is based purely on physical and accuracy considerations typical applications run on courant numbers ranging from 0 1 to 10 12 the performance of this numerical scheme is demonstrated on a planar 1 d shocktube and the formation of a one solar mass protostar in spherical symmetry

unified approach to simulation on deforming elements with application to phase change problems abstract several different approaches to simulation on deforming finite elements are shown to generate the same weighted residuals formulation for the evolution of the dependent variables control of node motion by means of mesh stretching in two dimensions is illustrated in the context of phase change problems simulation of problems with large phase boundary excursions shows good agreement with analytic solutions

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

a note on the grid movement induced by mfe moving grid methods in one space dimension have become popular for solving several kinds of parabolic and hyperbolic partial differential equations pdes involving fine scale structures such as steep moving fronts and emerging steep layers pulses shocks etc in two space dimensions however application of moving grid methods is less trivial than in 1d for some methods e g those based on equidistributing principles it is not even clear how to extend the underlying grid selection procedure to 2d the moving finite element mfe method does not suffer from this drawback its mathematical extension to 2d is trivial however because of the intrinsic coupling between the discretization of the pde and the grid selection the application of mfe as for any other method is not without difficulties in this paper we describe the node movement induced by mfe for various pdes and we indicate some problems concerning the grid structure that can result from this movement

on high order denoising models and fast algorithms for vector valued images variational techniques for gray scale image denoising have been deeply investigated for many years however little research has been done for the vector valued denoising case and the very few existent works are all based on total variation regularization it is known that total variation models for denoising gray scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector valued models high order models on the contrary do not present staircasing in this paper we introduce three high order and curvature based denoising models for vector valued images their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided ams subject classifications 68u10 65f10 65k10

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

an evaluation of the gradient weighted moving finite element method in one space dimension abstract moving grid methods are becoming increasingly popular for solving several kinds of parabolic and hyperbolic partial differential equations involving fine scale structures such as steep moving fronts and emerging steep layers an interesting example of such a method is provided by the moving finite element mfe method a difficulty with mfe as with many other existing moving grid methods is the threat of grid distortion which can only be avoided by the use of penalty terms the involved parameter tuning is known to be very important not only to provide for a safe automatic grid point selection but also for efficiency in the time stepping process when compared with mfe the gradient weighted mfe gwmfe method has some promising properties to reduce the need of tuning to investigate to what extent gwm fe can be called robust reliable and effective for the automatic solution of time dependent pdes in one space dimension we have tested this method extensively on a set of five relevant example problems with various solution characteristics all tests have been carried out using the bdf time integrator spgear of the existing method of lines software package sprint

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

on polymer conformations in elongational flows we consider various models of polymer conformations using paths of gaussian processes such as brownian motion in each case the calculation of the law of the moment of inertia of a random polymer structure which is equivalent to the calculation of the partition function is reduced to the problem of finding the law of a certain quadratic functional of a gaussian process we present a new method for computing the laplace transforms of these quadratic functionals which exploit their special form via the ray knight theorem and which does not involve the classical method of eigenvalue expansions we apply the method to several simple examples then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory

grid modification for second order hyperbolic problems a family of galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts the new methodology involves different finite element discretizations at different time levels thus at any time level relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts the change of grid from time step to time step need not be continuous and the number of grid points at different time levels can be arbitrarily different the formulation is applicable to general second order hyperbolic equations stability results are proved and a priori error estimates are established for several boundary conditions our error estimates consist of three parts the time finite difference discretization error the spatial finite element discretization error and the error due to the projections of the approximated solution from old grids onto new grids

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

a new moving finite element method based on quadratic approximation functions moving finite element methods are adaptive gridding procedures especially designed for systems of partial differential equations whose solutions contain steep gradients a new moving finite element method based on quadratic approximation functions is presented both the theoretical and computational aspects are outlined performance of the method is illustrated with solutions to burgers equation the solution is accurate and remarkably smooth in the entire domain

an adaptive mesh refinement strategy for finite element solution of the elliptic problem abstract in this paper we develop an adaptive finite element method for elliptic problems first we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration for some norm h s then an a posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations based on this new error estimator the element wise refinement strategy in each subdomain is provided for a given tolerance a discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family four 2d numerical benchmark examples of different domains and two 3d numerical benchmark examples are tested to demonstrate the robustness of our method when possible our numerical results are also compared to corresponding results from existing methods all the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems