a finite element method for unstructured grid smoothing the finite element method is applied to grid smoothing for two dimensional planar geometry the coordinates of the grid nodes satisfy two quasi linear elliptic equations in the form of laplace equations in a riemann space by forming a dirichlet boundary value problem the proposed method is applicable to both structured and unstructured grids the riemannian metric acting as a driving force in the grid smoothing is computed iteratively beginning with the metric of the unsmoothed grid smoothing is achieved by computing the metric tensor on the dual mesh elements which incorporates the influence of neighbor elements numerical examples of this smoothing methodology demonstrating the efficiency of the proposed approach are presented

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

efficient nonlinear solvers for laplace beltrami smoothing of three dimensional unstructured grids the laplace beltrami system of nonlinear elliptic partial differential equations has utility in the generation of computational grids on complex and highly curved geometry discretization of this system using the finite element method accommodates unstructured grids but generates a large sparse ill conditioned system of nonlinear discrete equations the use of the laplace beltrami approach particularly in large scale applications has been limited by the scalability and efficiency of solvers this paper addresses this limitation by developing two nonlinear solvers based on the jacobian free newton krylov jfnk methodology a key feature of these methods is that the jacobian is not formed explicitly for use by the underlying linear solver iterative linear solvers such as the generalized minimal residual gmres method do not technically require the stand alone jacobian instead its action on a vector is approximated through two nonlinear function evaluations the preconditioning required by gmres is also discussed two different preconditioners are developed both of which employ existing algebraic multigrid amg methods further the most efficient preconditioner overall for the problems considered is based on a picard linearization numerical examples demonstrate that these solvers are significantly faster than a standard newton krylov approach a speedup factor of approximately 26 was obtained for the picard preconditioner on the largest grids studied here in addition these jfnk solvers exhibit good algorithmic scaling with increasing grid size

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

uniform asymptotic approximations for incomplete riemann zeta functions an incomplete riemann zeta function z1 x is examined along with a complementary incomplete riemann zeta function z2 x these functions are defined by z1 x 1 21 1 0x t 1 et 1 1 dt and z2 x z1 x where is the classical riemann zeta function z1 x has the property that limx z1 x for re 0 and 1 the asymptotic behaviour of z1 x and z2 x is studied for the case re 0 fixed and im and using liouville green wkbj analysis asymptotic approximations are obtained complete with explicit error bounds which are uniformly valid for 0 x

quasi orthogonal grids with impedance matching an elliptic quasi orthogonal grid generation system is formulated based on quasi conformal mapping for arbitrary anisotropic long and skinny regions the resulting system is a generalization of the well known elliptic grid generation system derived from conformal mapping coupled with the grid generation system the impedance matching principle describes a methodology for preserving the discrete accuracy of the simulation both internal to the domain and near internal geometric interfaces empirically satisfying the impedance principle tends to minimize mesh effects on the solution results meshes that are impedance matched tend to reduce or eliminate spurious wave reflection and or attenuation at internal interfaces the resulting grid generation system is used to construct impedance matched quasi orthogonal grids on domains containing internal geometric constraints given an algebraic grid and a grid impedance function as initial conditions

existence of global bv solutions to a model of reacting euler fluid with variable thermodynamics parameters abstract we investigate the cauchy problems of euler equations coupled with chemical reaction equation the specific heat and adiabatic parameters of compressible fluid are introduced as functions with respect to reactant mass fraction in this paper by means of fractional step wave front tracking scheme we derive the existence of global bv solution and verify that it is indeed an entropy weak solution this system is also regarded as balance law with resonant characteristic fields hence some decay condition is imposed on the source term in order to handle the damping effect

singular perturbations of first order hyperbolic systems with stiff source terms abstract this work develops a singular perturbation theory for initial value problems of nonlinear first order hyperbolic systems with stiff source terms in several space variables it is observed that under reasonable assumptions many equations of classical physics of that type admit a structural stability condition this condition is equivalent to the well known subcharacteristic condition for one dimensional 2 2 systems and the well known time like condition for one dimensional scalar second order hyperbolic equations with a small positive parameter multiplying the highest derivatives under this stability condition we construct formal asymptotic approximations of the initial layer solution to the nonlinear problem furthermore assuming some regularity of the solutions to the limiting inner problem and the reduced problem we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval the stability condition seems to be a key to problems of this kind and can be easily verified moreover this presentation unifies and improves earlier works for some specific equations

local optimization based simplicial mesh untangling and improvement we present an optimization based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh these functions are linear with respect to the free vertex position thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method we prove that the function level sets are convex regardless of the position of the free vertex and hence the local subproblem is guaranteed to converge maximizing the minimum area or volume of mesh elements although well suited for mesh untangling is not ideal for mesh improvement and its use often results in poor quality meshes we therefore combine the mesh untangling technique with optimization based mesh improvement techniques and expand previous results to show that a commonly used two dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both two and three dimensional simplicial meshes

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

bv solutions for hyperbolic systems of balance laws with relaxation abstract for hyperbolic systems of balance laws with source manifesting relaxation it is shown that the kawashima condition which yields global classical solutions with smooth initial values near equilibrium is also instrumental in inducing the existence of global admissible bv solutions accommodating shocks

the slow erosion limit in a model of granular flow we study a 2 2 system of balance laws that describes the evolution of a granular material avalanche flowing downhill the original model was proposed by hadeler and kuttler granul matter 2 9 18 1999 the cauchy problem for this system has been studied by the authors in recent papers amadori and shen in commun partial differ equ 34 1003 1040 2009 shen in j math anal appl 339 828 838 2008 in this paper we first consider an initial boundary value problem the boundary condition is given by the flow of the incoming material for this problem we prove the global existence of bv solutions for a suitable class of data with bounded but possibly large total variations we then study the slow erosion or deposition limit we show that if the thickness of the moving layer remains small then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill not on the time law describing the rate at which the material slides down more precisely in the limit as the thickness of the moving layer tends to zero the slope of the mountain is provided by an entropy solution to a scalar integro differential conservation law

sparsity and its applications this volume is based on a series of lectures prepared by a group of internationally renowned experts in the field for a meeting on the general theme of sparsity and its applications at the university of technology loughborough in april 1983 an introductory section briefly surveys the field of sparsity research and summarises the main ideas discussed in each contribution the contributions themselves examine the design analysis implementation and application of computer algorithms with a view to producing mathematical software which can be used as an efficent tool in different areas of science and engineering technology

computation of a general integral of fermi dirac distribution by mcdougall stoner method abstract we extended the method of mcdougall and stoner 1938 15 to the computation of a general integral of the fermi dirac distribution f when 0 the new method splits f into a sum of three parts a b and c and integrates them exactly and or numerically by means of the double exponential quadrature rules as increases b f damps algebraically while c f decays exponentially thus they can be ignored when exceeds certain threshold values depending on the input error tolerance when a is exactly computable and is sufficiently large such that b and c are negligible the new method was 60 1000 times faster than the numerical integration of f even if b and or c are not negligible their smallness and rapid decay significantly accelerate their numerical quadrature so that the speed up factor becomes 3 20 unless is less than 2 0 4 5 on the other hand when a is numerically integrated the speed up factor diminishes to 2 5 still the superiority of the new method to the numerical integration of f is unchanged

hyperbolic systems of balance laws via vanishing viscosity abstract global weak solutions of a strictly hyperbolic system of balance laws in one space dimension are constructed by the vanishing viscosity method of bianchini and bressan for global existence a suitable dissipativeness assumption has to be made on the production term g under this hypothesis the viscous approximations u that are globally defined solutions to u t a u u x g u u xx satisfy uniform bv bounds exponentially decaying in time furthermore they are stable in l 1 with respect to the initial data finally as 0 u converges in l loc 1 to the admissible weak solution u of the system of balance laws u t f u x g u 0 when a df

gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation we study gas flow in vibrational nonequilibrium the model is a 4 4 nonlinear hyperbolic system with relaxation under physical assumptions properties of thermodynamic variables relevant to stability are obtained global existence for cauchy problems with smooth and small data is established and large time behavior is studied in the pointwise sense we formulate the fundamental solution in a systematic way for a general linear system with relaxation the fundamental solution provides insights to the behavior of the nonlinear system and is crucial to obtain our pointwise asymptotic picture for the nonequilibrium flow we also clarify in a general setting the relation between subcharacteristic conditions and a dissipative criterion that was originally proposed for hyperbolic parabolic systems and has now proved to be important also for hyperbolic systems with relaxation

systems of hyperbolic conservation laws with a resonant moving source in this paper we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source compared with the previous results w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 t p liu nonlinear stability and instability of transonic flows through a nozzle comm math phys 83 2 1982 243 260 on this stability problem in this paper the transonic ith shock is assumed to be relatively strong and stable in the sense of majda then the framework of m lewicka l1 stability of patterns of non interacting large shock waves indiana univ math j 49 4 2000 1515 1537 m lewicka stability conditions for patterns of noninteracting large shock waves siam j math anal 32 5 2001 1094 1116 electronic can be applied a new criterion is obtained to test whether such a shock is time asymptotically stable or not and by constructing the liu yang functional one can prove the l1 stability of the shock under the stability condition this is an extension of the result s y ha t yang l1 stability for systems of hyperbolic conservation laws with a resonant moving source siam j math anal 34 5 2003 1226 1251 electronic w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 to a more general case

analysis of the m1 model well posedness and diffusion asymptotics abstract this paper is devoted to the analysis of the m 1 model which arises in radiative transfer theory the derivation of the model is based on the entropy minimization principle which leads to a hyperbolic system of balance laws with relaxation in the multi dimensional case we establish the existence uniqueness of a globally defined smooth solution under a suitable smallness condition on the initial data in the one dimensional case we show that the smallness condition does not depend on the particles mean free path so that we can also rigorously justify the consistency of the model with the diffusion asymptotics the result extends the analysis of coulombel et al j f coulombel f golse t goudon diffusion approximation and entropy based moment closure for kinetic equations asymptotic analysis 45 2005 1 39 to the case where the entropy functional accounts for relaxation towards the planckian state which is physically more relevant

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

conditional symmetries and riemann invariants for inhomogeneous hydrodynamic type systems a new approach to the solution of quasilinear nonelliptic first order systems of inhomogeneous partial differential equations in many dimensions is presented it is based on a version of the conditional symmetry and riemann invariant methods we discuss in detail the necessary and sufficient conditions for the existence of rank 2 and rank 3 solutions expressible in terms of riemann invariants we perform the analysis using the cayley hamilton theorem for a certain algebraic system associated with the initial system the problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of riemann invariants a couple of theorems useful for the construction of such solutions are given these theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and coriolis forces several new rank 2 solutions are obtained some physical interpretation of these results is given

sharp decay estimates for hyperbolic balance laws abstract we consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one space dimension sharp decay estimates are derived for the positive waves in an entropy weak solution the result is obtained by introducing a partial ordering within the family of positive radon measures using symmetric rearrangements and a comparison with a solution of burgers s equation with impulsive sources as well as lower semicontinuity properties of continuous glimm type functionals

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

uniform global existence and parabolic limit for partially dissipative hyperbolic systems abstract this work concerns smooth solutions to the cauchy problem for first order partially dissipative hyperbolic systems with a small parameter the systems are written in non conservative form in several space variables we introduce algebraic conditions on the structure of the systems under these conditions together with a partial dissipation condition and the shizuta kawashima stability condition we prove three main results around constant equilibrium states these results are uniform global existence with respect to the parameter global in time convergence of the systems to second order nonlinear parabolic systems in a slow time variable and global existence when the parameter is fixed we also give examples of physical models to which the above results can be applied

existence of relaxation shock profiles for hyperbolic conservation laws in this paper we prove that like viscosity relaxation can also smooth weak shocks for strict hyperbolic conservation laws as equilibium systems of hyperbolic relaxation systems under some natural structural conditions these conditions were derived previously by yong from stability considerations and hence are entirely analogous to the majda pego criterion of the viscous case the proof involves a new parametrization of hugoniot curves and a delicate analysis of the algebraic structure of relaxation systems as well as construction of an appropriate center manifold

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

validity of the chapman enskog expansion for a class of hyperbolic relaxation systems this paper presents a validity proof of the chapman enskog expansion for a class of hyperbolic relaxation systems this class is characterized with a set of conservation dissipation structural conditions and contains many classical models from mathematical physics our main result confirms that the hyperbolic relaxation systems can be well approximated in the time interval where the corresponding conservation laws have smooth solutions with the corresponding viscous systems derived by using the chapman enskog expansion

a finite element method for three dimensional unstructured grid smoothing the finite element method is applied to grid smoothing in three dimensional geometry generalizing earlier results obtained for planar geometry the underlying set of equations for the cartesian components of grid coordinates based on the notion of harmonic coordinates has a natural variational formulation to estimate the target metric tensor that drives the elliptic grid equations the metric tensor components are computed on a coarse grained grid numerical examples illustrating the proposed approach are presented together with results from the smoothness functional which is used to measure the quality of the resulting grid

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

hyperbolic principal subsystems entropy convexity and subcharacteristic conditions we consider a system of n balance laws compatible with an entropy principle and convex entropy density using the special symmetric form induced by the main field we define the concept of principal subsystem associated with the system we prove that the 2 n 2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law moreover we can verify that for each principal subsystem the maximum minimum characteristic velocity is not larger smaller than the maximum minimum characteristic velocity of the full system these are the subcharacteristic conditions we present some simple examples in the case of the euler fluid then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics finally in the moments approach to the boltzmann equation we prove as a consequence of the previous result that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

unstructured surface mesh adaptation using the laplace beltrami target metric approach this paper develops a set of adaptive surface mesh equations by using a harmonic morphism which is a special case of a harmonic map these equations are applicable both to structured and unstructured surface meshes provided that the underlying surface is given in a parametric form by representing the target metric of the mesh as a sum of a coarse grained component and a component quadratic in surface gradients an improved surface mesh may be obtained the weak form of the grid equations is solved using the finite element approximation which reduces the grid equations to a nonlinear algebraic set examples of structured and unstructured meshes are used to illustrate the applicability of the proposed approach

global solutions with shock waves to the generalized riemann problem for a system of hyperbolic conservation laws with linear damping abstract it is proven that a class of the generalized riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise c 1 solution u u t x containing only n shock waves with small amplitude on t 0 and this solution possesses a global structure similar to that of the similarity solution u u x t of the corresponding homogeneous riemann problem as an application of our result we prove the existence of global shock solutions piecewise continuous and piecewise smooth solution with shock discontinuities of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data we also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves

mesh relaxation a new technique for improving triangulations summary given a list of points defining a domain boundary a three stage process is often used to triangulate a domain first an appropriate distribution of interior points is generated next the points are connected to form triangles and finally the connectivity data are used to reposition the interior points using the laplacian smoothing technique thereby usually improving the shapes of the triangles this paper describes a new technique for mesh improvement adjusting the connection structure during the second stage of this process the new scheme which we call mesh relaxation consists of a procedure for iteratively making the mesh topology more regular by edge swapping for each interior edge a relaxation index is computed that depends on the degrees of its end points and adjacent points any edge for which this index exceeds a prescribed threshold will be swapped i e replaced by a new edge connecting the adjacent points of the original edge after all edge swaps are completed laplacian smoothing is applied to the mesh examples show that when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

computational conformal mapping for surface grid generation the paper describes the development and application of a new approach for formulating an elliptic generation system on parametrically defined surfaces the present derivation of the surface equations proceeds in two steps first conformal mapping of smooth surfaces onto rectangular regions is utilized to derive a first order system of partial differential equations analogous to beltrami s system for quasi conformal mapping of planar regions second a general elliptic generation system for three dimensional surfaces including forcing functions is formulated based on beltrami s system and quasi conformal mapping the resulting elliptic system is solved using an iterative method on arbitrary surfaces represented analytically by rational b splines the overall effect of this approach is a reliable and versatile elliptic method for generating and improving surface grids examples will be presented to demonstrate the application of the method in constructing practical grids

elliptic grid generation based on laplace equations and algebraic transformations abstract an elliptic grid generation method is presented to generate boundary conforming grids in domains in 2d and 3d physical space and on minimal surfaces and parametrized surfaces in 3d physical space the elliptic grid generation method is based on the use of a composite mapping this composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation the elliptic transformation is based on the laplace equations for domains or on the laplace beltrami equations for surfaces the algebraic transformation maps the computational space one to one onto a parameter space the elliptic transformation maps the parameter space one to one onto the domains or surfaces the composition of these two mappings is a differentiable one to one mapping from computational space onto the domains or surfaces and has a nonvanishing jacobian this composite mapping defines the grid point distribution in the interior of the domains or surfaces for domains and minimal surfaces the composite mapping obeys a nonlinear elliptic poisson system with control functions completely defined by the algebraic transformation the solution of the poisson systems is obtained by picard iteration and black box multigrid solvers for parametrized curved surfaces it is not necessary to define and solve a nonlinear elliptic poisson system instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface

multilevel first order system least squares for elliptic grid generation a new fully variational approach is studied for elliptic grid generation egg it is based on a general algorithm developed in a companion paper a l codd t a manteuffel and s f mccormick siam j numer anal 41 2003 pp 2197 2209 that involves using newton s method to linearize an appropriate equivalent first order system first order system least squares fosls to formulate and discretize the newton step and algebraic multigrid amg to solve the resulting matrix equation the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation the present paper verifies the assumptions of the companion work and confirms the overall efficiency of the scheme with numerical experiments

bv weak solutions to gauss codazzi system for isometric immersions abstract the isometric immersion problem for surfaces embedded into r 3 is studied via the fluid dynamic framework introduced in chen et al 2010 6 as a system of balance laws of mixed type the techniques developed in the theory of weak solutions of bounded variation in continuum physics are employed to deal with the isometric immersions in the setting of differential geometry the so called bv framework is formed that establishes convergence of approximate solutions of bounded variation to the gauss codazzi system and yields the c 1 1 isometric realization of two dimensional surfaces into r 3 local and global existence results are established for weak solutions of small bounded variation to the gauss codazzi system for negatively curved surfaces that admit equilibrium configurations as an application the case of catenoidal shell of revolution is provided

entropy and global existence for hyperbolic balance laws this paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables we propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state in addition we show that a system of balance laws satisfies a kawashima condition if and only if its first order approximation namely the hyperbolic parabolic system derived through the chapman enskog expansion satisfies the corresponding kawashima condition the result is then applied to bouchut s discrete velocity bgk models approximating hyperbolic systems of conservation laws

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t