the classical limit of quantum mechanics as a lie algebra contraction the classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non compact kinematical algebra w1 the weyl heisenberg algebra to the three dimensional abelian algebra t3 an appropriate definition of the contraction of a lie algebra and of a sequence of its representations is given for some quantum systems with simple explicitly integrable dynamics it is shown how the classical poisson bracket and classical trajectories are obtained in the limit each trajectory is associated with a one dimensional representation of t3 within a direct integral of such representations in a hilbert space each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics and a superselection rule arises naturally between any two such subspaces

a posteriori error estimates for combined finite volume finite element discretizations of reactive transport equations on nonmatching grids abstract we derive in this paper guaranteed and fully computable a posteriori error estimates for vertex centered finite volume type discretizations of transient linear convection diffusion reaction equations our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew symmetric part of the differential operator lower bounds global in space but local in time are also derived these lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting on the basis of the derived estimates we propose an adaptive algorithm which enables to automatically achieve a user given relative precision this algorithm also leads to efficient calculations as it balances the time and space error contributions as an example we apply our estimates to the combined finite volume finite element scheme including such features as use of mass lumping for the time evolution or reaction terms of upwind weighting for the convection term and discretization on nonmatching meshes possibly containing nonconvex and non star shaped elements a collection of numerical experiments illustrates the efficiency of our estimates and the use of the space time adaptive algorithm

variational multiscale element free galerkin vmefg and local discontinuous galerkin ldg methods for solving two dimensional brusselator reaction diffusion system with and without cross diffusion abstract the finite element method fem is one of the basic methods for solving deterministic and stochastic partial differential equations this method is proposed in the 19 decade and after years several modifications for this well known technique such as mortal fem discontinuous galerkin fem extended fem least squares fem spectral fem mixed fem immersed fem adaptive fem etc have been proposed also for improving the accuracy and for considering complex domain shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique the name of this improved method is element free galerkin efg method which is classified in the category of meshless methods the efg method has been applied for solving many problems in engineering in the current paper we select two numerical procedures that are extracted from fem the discontinuous galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity it should be noted that many enriched ideas have been explained for improving accuracy of efg method in the current paper the local discontinuous galerkin and variational multiscale efg methods are applied for obtaining numerical solution of two dimensional brusselator system with and without cross diffusion the brusselator system is a theoretical model for a type of autocatalytic reaction up to best of authors knowledge the accuracy of the obtained numerical solutions using local discontinuous galerkin method is very related to the used time discrete scheme therefore fourth order exponential time differencing method has been employed for discretizing the time variable also to achieve a full discretization scheme the local discontinuous galerkin finite element method is used moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

convergence properties of the runge kutta chebyshev method the runge kutta chebyshev method is ans stage runge kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations method of lines the method possesses an extended real stability interval with a length proportional tos 2 the method can be applied withs arbitrarily large which is an attractive feature due to the proportionality of withs 2 the involved stability property here is internal stability internal stability has to do with the propagation of errors over the stages within one single integration step this internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space time grid refinement for a model class of linear problems we prove convergence under the sole condition that the necessary time step restriction for stability is satisfied these error bounds are valid for anys and independent of the stiffness of the problem numerical examples are given to illustrate the theoretical results

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

lattice boltzmann simulation of pattern formation under cross diffusion in this paper a lattice boltzmann model for cross reaction diffusion system is proposed in order to recover the cross diffusion equations by lattice boltzmann method we transform the equations into a complex reaction diffusion equation this complex reaction diffusion equation is recovered with higher order accuracy of the truncation error based on this model two typical reaction diffusion systems with cross diffusion predator prey system and inhomogeneous brusselator model are simulated the numerical results show that this model can be used to simulate the cross reaction diffusion systems

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

smooth macro elements based on clough tocher triangle splits summary macro elements of smoothness c r on clough tocher triangle splits are constructed for all r ge 0 these new elements are improvements on elements constructed in 11 in that disproving a conjecture made there certain unneeded degrees of freedom have been removed numerical experiments on hermite interpolation with the new elements are included

accidental degeneracy and symmetry lie algebra the authors consider a very elementary example of accidental degeneracy and show that it is not explained by the symmetry lie algebra of the corresponding classical problem

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems an alternating direction implicit adi orthogonal spline collocation osc method is described for the approximate solution of a class of nonlinear reaction diffusion systems its efficacy is demonstrated on the solution of well known examples of such systems specifically the brusselator gray scott gierer meinhardt and schnakenberg models and comparisons are made with other numerical techniques considered in the literature the new adi method is based on an extrapolated crank nicolson osc method and is algebraically linear it is efficient requiring at each time level only o n operations where n is the number of unknowns moreover it is shown to produce approximations which are of optimal global accuracy in various norms and to possess superconvergence properties

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

a stabilized finite volume element formulation for sedimentation consolidation processes a model of sedimentation consolidation processes in so called clarifier thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the stokes system for an incompressible fluid describing the motion of the mixture in cylindrical coordinates and if an axially symmetric solution is assumed the original problem reduces to two space dimensions this poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted sobolev spaces a novel finite volume element method is introduced for the spatial discretization where the velocity field and the solids concentration are discretized on two different dual meshes the method is based on a stabilized discontinuous galerkin formulation for the concentration field and a multiscale stabilized pair of mathbb p 1 mathbb p 1 elements for velocity and pressure respectively numerical experiments illustrate properties of the model and the satisfactory p

double and triple points in the spectra of the cubic symmetry hamiltonian accidental degeneracy of levels is studied in the spectrum of the hamiltonian used frequently in applications which is a linear combination of spherical tensors rank n or 6 invariant with respect to transformations of a cubic group the degeneracy arises only at definite values of the hamiltonian parameters and seems to be irrelevant with respect to increasing hamiltonian symmetry the basis of irreducible representations dgamma of the cubic group is constructed in the space of irreducible representations dj of the so 3 group

weakly nonlinear analysis of turing patterns in a morphochemical model for metal growth we focus on the morphochemical reaction diffusion model introduced in bozzini et al 2013 and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of turing instability of the physically relevant equilibrium we perform a weakly nonlinear multiple scales analysis and derive the normal form equations governing the amplitude of the patterns these amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations hysteretic type phenomena are highlighted also through numerical simulations we show the occurrence of spatial pattern propagation and derive the ginzburg landau equation describing the envelope of the traveling wavefront

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

operator splitting for nonlinear reaction diffusion systems with an entropic structure singular perturbation and order reduction in this paper we perform the numerical analysis of operator splitting techniques for nonlinear reaction diffusion systems with an entropic structure in the presence of fast scales in the reaction term we consider both linear diagonal and quasi linear non diagonal diffusion the entropic structure implies the well posedness and stability of the system as well as a tikhonov normal form for the nonlinear reaction term 23 it allows to perform a singular perturbation analysis and to obtain a reduced and well posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution we then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step t thus leading to the failure of the usual splitting analysis techniques we define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales all the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level we finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis

on super splines and finite elements super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges dimension formulae for super spline spaces are established and the construction of explicit bases of locally supported splines is presented in addition it is shown that many classical finite element spaces can be interpreted as super spline spaces providing a link between spline theory and the highly applicable finite element theory

finite element approximation of a nonlinear cross diffusion population model we consider a fully discrete finite element approximation of the nonlinear cross diffusion population model find ui the population of the ith species i 1 and 2 such that where j i and gi u1 u2 i ii ui ij uj ui in the above the given data is as follows v is an environmental potential ci ai are diffusion coefficients bi are transport coefficients i are the intrinsic growth rates and ii are intra specific whereas ij i j are interspecific competition coefficients in addition to showing well posedness of our approximation we prove convergence in space dimensions d 3 finally some numerical experiments in one space dimension are presented

a new dimension to turing patterns abstract it is well known that simple reaction diffusion systems can display very rich pattern formation behaviour here we have studied two examples of such systems in three dimensions first we investigate the morphology and stability of a generic turing system in three dimensions and then the well known gray scott model in the latter case we added a small number of morphogen sources in the system in order to study its robustness and the formation of connections between the sources our results raise the question of whether turing patterning can produce an inductive signalling mechanism for neuronal growth

c1 quintic interpolation over triangles two explicit representations two explicit representation of a c1 quintic interpolant over triangles are given these representations are derived by generalization of coons methods and bernstein bezier methods respectively