further results on least squares based adaptive minimum variance control based on the recently established results on self tuning regulators originally proposed by astrom and wittenmark this paper presents various novel and extended results on least squares based adaptive minimum variance control for linear stochastic systems these results establish self optimality self tuning property and the best possible convergence rate of the control law in a variety of situations of interest

computationally efficient adaptive time step method for the cahnhilliard equation in this work we propose a fast and efficient adaptive time step procedure for the cahnhilliard equation the temporal evolution of the cahnhilliard equation has multiple time scales for spinodal decomposition simulation an initial random perturbation evolves on a fast time scale and later coarsening evolves on a very slow time scale therefore if a small time step is used to capture the fast dynamics the computation is quite costly on the other hand if a large time step is used fast time evolutions may be missed hence it is essential to use an adaptive time step method to simulate phenomena with multiple time scales the proposed time adaptivity algorithm is based on the discrete maximum norm of the difference between two consecutive time step numerical solutions numerical experiments in one two and three dimensions are presented to demonstrate the performance and effectiveness of the adaptive time stepping algorithm

separation of variables for the generalized henon heiles system and system with quartic potential we consider two well known integrable systems on the plane using the concept of natural poisson bivectors on riemannian manifolds a geometric approach to construction of variables of separation and separated relations for the generalized henon heiles system and the generalized system with quartic potential is discussed in detail

stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation in this paper we present and compare two unconditionally energy stable finite difference schemes for the phase field crystal equation the first is a one step scheme based on a convex splitting of a discrete energy by wise et al s m wise c wang j s lowengrub an energy stable and convergent finite difference scheme for the phase field crystal equation siam j numer anal in press in this scheme which is first order in time and second order in space the discrete energy is non increasing for any time step the second scheme we consider is a new fully second order two step algorithm in the new scheme the discrete energy is bounded by its initial value for any time step in both methods the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable regardless of time step size we solve the nonlinear equations using an efficient nonlinear multigrid method numerical simulations are presented and confirm the stability efficiency and accuracy of the schemes

quasi point separation of variables for the henon heiles system and a system with a quartic potential we examine the problem of integrability of two dimensional hamiltonian systems by means of separation of variables a systematic approach to the construction of the special non pure coordinate separation of variables for certain natural two dimensional hamiltonians is proposed

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

dynamic solid solid transitions with phase characterized by an order parameter abstract we develop a thermodynamically consistent continuum theory for the study of solid solid phase transitions where deformation dominates heat and mass transfer we use an order parameter to characterize the notion of phase and identify phase interfaces with thin transition zones within which the strain and order parameter exhibit large gradients to model the growth of one phase at the expense of another we introduce configurational forces subject to their own balance that work against changes in the order parameter by studying traveling waves and the asymptotics of a transition layer we establish connections between our theory and more standard approaches that identify phase interfaces with surfaces of strain discontinuity

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

a positivity preserving finite element method for chemotaxis problems in 3d we present an implicit finite element method for a class of chemotaxis models in three spatial dimensions the proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density to enforce the discrete maximum principle the standard galerkin discretization is constrained using a local extremum diminishing flux limiter to demonstrate the efficiency and robustness of this approach we solve blow up problems in a 3d chemostat domain to give a flavor of more complex and realistic chemotactic applications we investigate the pattern dynamics and aggregating behavior of the bacteria escherichia coli and salmonella typhimurium the obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

almost sure rate of convergence of the parameter estimates in stochastic approximation algorithm this paper presents novel results on the almost sure convergence of the parameter estimates in stochastic approximation algorithm it is proved that this rate has the same order as the best one established for the least squares algorithm although we consider the case of self tuning controllers the presented results can easily be extended to some other adaptive processes

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model we develop a family of new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model this model is described by a system of two nonlinear pdes a convection diffusion equation for the cell density coupled with a reaction diffusion equation for the chemoattractant concentration it has been recently shown that the convective part of this system is of a mixed hyperbolic elliptic type which may cause severe instabilities when the studied system is solved by straightforward numerical methods therefore the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original keller segel model in the form of a convection diffusion reaction system with a hyperbolic convective part we then design interior penalty discontinuous galerkin methods for the rewritten keller segel system our methods employ the central upwind numerical fluxes originally developed in the context of finite volume methods for hyperbolic systems of conservation laws in this paper we consider cartesian grids and prove error estimates for the proposed high order discontinuous galerkin methods our proof is valid for pre blow up times since we assume boundedness of the exact solution we also show that the blow up time of the exact solution is bounded from above by the blow up time of our numerical solution in the numerical tests presented below we demonstrate that the obtained numerical solutions have no negative values and are oscillation free even though no slope limiting technique has been implemented

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

the periodic isoperimetric problem given a discrete group g of isornetries of r 3 we study the g isoperimetric problem which consists of minimizing area modulo g among surfaces in r 3 which enclose a g invariant region with a prescribed volume fraction if g is a line group we prove that solutions are either families of round spheres or right cylinders in the doubly periodic case we prove that for most rank two lattices solutions must be spheres cylinders or planes for the remaining rank two lattices we show among other results an isoperimetric inequality in terms of the topology of the isoperimetric surfaces finally we study the case where g pm3m the group of symmetries of the integer rank three lattice z 3 and other crystallographic groups of cubic type we prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than 1 6 and we give an isoperimetric inequality for g invariant regions that for instance implies that the area modulo z 3 of a surface dividing the three space in two g invariant regions with equal volume fractions is at least 2 19 the conjectured solution is the classical p schwarz triply periodic minimal surface whose area is 2 34 another consequence of this isoperimetric inequality is that pm3m symmetric surfaces other than families of spheres cannot be isoperimetric for the lattice group z 3

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

lax representation for two particle dynamics splitting on two tori the lax representation in terms of matrices is constructed for separable multiply periodic systems splitting on two tori hyperelliptic kleinian functions and their reduction to elliptic functions are used

solving a chemotaxis haptotaxis system in 2d using generalized finite difference method abstract we study a mathematical model of cancer cell invasion of tissue extracellular matrix consisting of a system of reaction diffusion taxis partial differential equations which describes the interactions between cancer cells the matrix degrading enzyme and the host tissue we analyze the local stability of the constant equilibrium solutions to the chemotaxis haptotaxis system we derive a discretization of the system by means of the generalized finite difference method gfdm and we prove the convergence of the discrete solution to the analytical one also we provide several numerical examples on the applications of this meshless method over regular and irregular domains

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

the zero divisor problem of multivariable stochastic adaptive control abstract stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time in order that the analysis of these algorithms is valid zero divisions must be events of probability zero this property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to lebesgue measure this result is deduced from the following general result established in this paper a non constant rational function of a finite set of random variables x 1 x n is absolutely continuous with respect to lebesgue measure if the joint distribution function of x 1 x n has this property

numerical study of three dimensional turing patterns using a meshless method based on moving kriging element free galerkin efg approach in this paper a numerical procedure is presented for solving a class of three dimensional turing system first we discrete the spatial direction using element free galerkin efg method based on the shape functions of moving kriging interpolation then to achieve a high order accuracy we use the fourth order exponential time differencing runge kutta etdrk4 method using this discretization for the temporal dimension we obtain an explicit scheme and do not need to solve nonlinear system of equations the efg method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the efg method test and trial functions are moving least squares approximation mls shape functions since the shape functions of moving least squares mls approximation do not have kronecker delta property we cannot implement the essential boundary condition directly also building shape functions of mls approximation is a time consuming procedure because of the mentioned reasons we employ the shape functions of moving kriging interpolation technique which have the mentioned property and less cpu time is required for building them for testing this method on three dimensional pdes we select some equations and system of pdes such as allen cahn gray scott ginzburg landau brusselator models predator prey model with additional food supply to predator several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

galerkin finite element method for cancer invasion mathematical model abstract a finite element scheme for the solution of a cancer invasion model is proposed the cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density extra cellular matrix and the matrix degrading enzymes the model incorporates proliferation and haptotaxis effect of cancer cells their interaction with extracellular matrix the production of matrix degrading enzymes and consequent degradation of the extracellular matrix the coupled partial differential equations are discretized in space with the standard galerkin finite elements and in time with the crank nicolson method moreover the nonlinear terms in the coupled equations are treated semi implicitly in the finite element scheme the numerical scheme is validated with numerical results taken from the literature in addition to the mesh convergence study the effects of haptotactic rate proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated

nonnegativity of exact and numerical solutions of some chemotactic models we investigate nonnegativity of exact and numerical solutions to a generalized keller segel model this model includes the so called minimal keller segel model but can cover more general chemistry we use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative we then derive numerical methods using finite element techniques for the generalized keller segel model adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions one of the numerical methods is then applied to the minimal keller segel model recalling known results on the qualitative behavior of this model we are able to choose parameters that yield convergence to a nonhomogeneous stationary solution while proceeding to exhibit these stationary patterns we also demonstrate how naive choices of numerical methods can give physically unrealistic solutions thereby justifying the need to study positivity preserving methods

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

operator splitting combined with positivity preserving discontinuous galerkin method for the chemotaxis model the advection diffusion reaction adr systems are often used to describe the chemotaxis models arising in biology in general the stiffness from the reaction and diffusion terms often requires very restricted time step size and the advection term depending on the concentration gradients of another component the chemoattractant may lead to sharp peaks in localized spatial regions it is challenging to design numerical methods that can efficiently handle both difficulties in this paper we apply the operator splitting approach to solve the advection diffusion reaction systems in particular for advection term we use the positivity preserving dg method with strong stability preserving ssp high order time discretizations for reaction diffusion term direct discontinuous galerkin ddg method is used in spatial discretization and krylov iif method is applied in time discretization numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method

computationally efficient solution to the cahn hilliard equation adaptive implicit time schemes mesh sensitivity analysis and the 3d isoperimetric problem we present an efficient numerical framework for analyzing spinodal decomposition described by the cahn hilliard equation we focus on the analysis of various implicit time schemes for two and three dimensional problems we demonstrate that significant computational gains can be obtained by applying embedded higher order runge kutta methods in a time adaptive setting this allows accessing time scales that vary by five orders of magnitude in addition we also formulate a set of test problems that isolate each of the sub processes involved in spinodal decomposition interface creation and bulky phase coarsening we analyze the error fluctuations using these test problems on the split form of the cahn hilliard equation solved using the finite element method with basis functions of different orders any scheme that ensures at least four elements per interface satisfactorily captures both sub processes our findings show that linear basis functions have superior error to cost properties this strategy coupled with a domain decomposition based parallel implementation let us notably augment the efficiency of a numerical cahn hillard solver and open new venues for its practical applications especially when three dimensional problems are considered we use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions the framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3d sphere cylinder lamella doubly periodic surface with genus two lawson surface and triply periodic minimal surface p schwarz surface

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

efficient accurate and flexible finite element solvers for chemotaxis problems in the framework of finite element discretizations we introduce a fully nonlinear newton like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities a chemical agent and the reacting population of certain biological organisms species we investigate the benefit of a corresponding monolithic approach and the decoupled variant in particular we analyze accuracy efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems

discontinuous galerkin methods for the chemotaxis and haptotaxis models in this work first we formulate and compare three different discontinuous interior penalty galerkin methods for the 2d keller segel chemotaxis model keller segel chemotaxis model is the important starting step in the modeling of the real biological system we show in the numerical tests that two of the proposed methods fail to give accurate oscillation free solutions next we consider the application of the successful method for the keller segel model to the simulation of the more realistic and closely related haptotaxis model of tumor invasion into healthy tissues

numerical simulation of reaction diffusion neural dynamics models and their synchronization desynchronization application to epileptic seizures abstract cognitive disorders especially epilepsy are closely linked with synchronization desynchronization of neurons in the brain in this paper the dynamical modeling and behavior analysis of a fitzhugh nagumo neuron and also synchronization control of a network of fitzhugh nagumo neurons which promise the understanding of cognitive processing are studied to numerically simulate these models and in order to overcome their difficulties such as computational complexity multi dimensionality non linearity having large spatial and temporal domains and also having a discrete initial data we propose the use of a strongly meshless technique based on wendland compactly supported radial basis function in conjunction with an operator splitting algorithm the main advantage of the proposed algorithm is its computational complexity for example o n x 1 3 n x 2 3 of the inversion of all matrices which is much lower than the complexity of pure radial basis functions method o n x 1 3 n x 2 3 of the inversion of matrices numerical experiments are presented showing that the proposed approaches are extremely accurate and fast

self tuning control based on multi innovation stochastic gradient parameter estimation abstract this paper uses the multi innovation stochastic gradient misg algorithm to estimate the parameters of discrete time systems and presents an misg based self tuning control scheme furthermore we prove that the parameter estimation error converges to zero under persistent excitation and the parameter estimation based control algorithm can asymptotically achieve virtually optimal control and ensure that the closed loop systems are stable and globally convergent the simulation example is included

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

an afc stabilized implicit finite element method for partial differential equations on evolving in time surfaces in this article we present a new implicit numerical scheme for reaction diffusion advection equations on an evolving in time hypersurface t the partial differential equations are solved on a stationary quadrilateral resp hexahedral mesh the zero level set of the time dependent indicator function t implicitly describes the position of t the dominating convective like terms which are due to the presence of chemotaxis transport of the cell density and surface evolution may lead to the non positiveness of a given numerical scheme and in such a way cause appearance of negative values and give rise of nonphysical oscillations in the numerical solution the proposed finite element method is constructed to avoid this problem implicit treatment of corresponding discrete terms in combination with the algebraic flux correction afc techniques make it possible to obtain a sufficiently accurate solution for reaction diffusion advection pdes on evolving surfaces

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

a petrov galerkin finite element method for simulating chemotaxis models on stationary surfaces abstract in this paper we present a petrov galerkin finite element method for a class of chemotaxis models defined on surfaces which describe the movement by one community in reaction to one chemical or biological signal on manifolds it is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property which is a challenge due to the singular behavior of numerical solution thus a petrov galerkin method is combined with an effective mass conservation factor to overcome the challenge furthermore we prove two facts this method maintains positivity and discrete mass conservation property in addition decoupled approach is applied based on the gradient and laplacian recoveries to solve the coupling system the relevant stability analyses is provided finally numerical simulations of blowing up problems and pattern formulations demonstrate the effectiveness of the proposed method

a multigrid finite element solver for the cahn hilliard equation a multigrid finite element solver for the cahn hilliard equation is presented that has mesh independent convergence rates for any time step size including in the important limit e 0 which is examined via numerical examples numerics are performed for a number of test problems which show that the features of the cahn hilliard equation minimising interface measure lyapunov energy functional etc are preserved we also explore the use of this solver in conjunction with adaptive time stepping and adaptive mesh strategies

numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method abstract in this paper the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces the models and the numerical method concerned are in terms of a generic form of chemotaxis models we develop a new semi implicit finite element scheme based on the gradient and laplacian recoveries to linearize the equations meanwhile we modify the proposed semi implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface the lumped mass modification is employed for positivity preservation and the related analysis results are provided we investigate the accuracy and convergence of the proposed method by numerical tests and the simulations of blowing up solution pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods

a positivity preserving high order discontinuous galerkin scheme for convection diffusion equations abstract for constructing high order accurate positivity preserving schemes for convection diffusion equations we construct a simple positivity preserving diffusion flux discontinuous galerkin dg schemes with such a positivity preserving diffusion flux are nonlinear schemes which can be regarded as a reduction of the high order positivity preserving dg schemes for compressible navier stokes equations in 1 to scalar diffusion operators in this paper we focus on the local dg method to discuss how to apply such a flux a limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity preserving in the sense that dg schemes with this flux satisfies a weak positivity property together with a positivity preserving limiter high order dg schemes with strong stability preserving time discretizations can be rendered positivity preserving without losing conservation or high order accuracy for convection diffusion problems with periodic boundary conditions or a special class of dirichlet or neumann boundary conditions numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme

universal dynamics of phase field models for dendritic growth we compare time dependent solutions of different phase field models for dendritic solidification in two dimensions including a thermodynamically consistent model and several ad hoc models the results are identical when the phase field equations are operating in their appropriate sharp interface limit the long time steady state results are all in agreement with solvability theory no computational advantage accrues from using a thermodynamically consistent phase field model

phase field model for three dimensional dendritic growth with fluid flow we study the effect of fluid flow on three dimensional 3d dendrite growth using a phase field model on an adaptive finite element grid in order to simulate 3d fluid flow we use an averaging method for the flow problem coupled to the phase field method and the semi implicit approximated projection method siapm we describe a parallel implementation for the algorithm using the charm fem framework and demonstrate its efficiency we introduce an improved method for extracting dendrite tip position and tip radius facilitating accurate comparison to theory we benchmark our results for 2d dendrite growth with solvability theory and previous results finding them to be in good agreement the physics of dendritic growth with fluid flow in three dimensions is very different from that in two dimensions and we discuss the origin of this behavior

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing

a plausible extension of standard penalty streamline upwind and immersed boundary techniques to the improved element free galerkin based solution of incompressible navier stokes equations abstract the present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element free galerkin iefg method for the numerical solution of incompressible fluid flow problems in principle the numerical procedures to be implemented in this communication have been conceived for finite element method fem based solutions and these include the reduced integration penalty method ripm the streamline upwind petrov galerkin supg scheme and a penalty based immersed boundary method pbibm for the imposition of essential boundary conditions along internal fluid solid interfaces the linear momentum balance and mass conservation equations have been coupled via the ripm in order to obtain a global weak formulation where the iefg model is entirely developed in terms of improved moving least squares imls approximations of the velocity field a detailed explanation concerning the appropriate extension of both the ripm and supg procedures to the context of iefg formulations has also been provided the resulting formulation has been applied to the solution of two well known benchmark problems i lid driven square cavity flow and ii flow past a fixed cylinder regarding the flow past a fixed cylinder benchmark problem the fluid solid interaction has been imposed as an internal immersed boundary condition via the pbibm the feasibility and reliability of implementing the ripm supg and pbibm procedures in the iefg formulation have been proven by comparison with experimental and mesh based numerical results reported in the literature the results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the iefg formulation allows the achievement of accurate and stable numerical results during the solution of incompressible fluid dynamics problems

finite element approximation of a phase field model for surface diffusion of voids in a stressed solid we consider a fully practical finite element approximation of the degenerate cahn hilliard equation with elasticity find the conserved order parameter theta x t in 1 1 and the displacement field underline u x t in bbb r 2 such that gamma frac partial theta partial t nabla cdot b theta nabla gamma delta theta gamma 1 psi theta tfrac 12 c theta cal c underline underline cal e underline u underline underline cal e underline u quad nabla cdot c theta cal c underline underline cal e underline u underline 0 subject to an initial condition theta 0 cdot in 1 1 on theta and boundary conditions on both equations here gamma in bbb r 0 is the interfacial parameter psi is a nonsmooth double well potential underline underline cal e is the symmetric strain tensor cal c is the possibly anisotropic elasticity tensor c s c 0 frac 12 1 c 0 1 s with c 0 gamma in bbb r 0 and b s 1 s 2 is the degenerate diffusion mobility in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions finally some numerical experiments are presented

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

a conservative numerical method for the cahn hilliard equation in complex domains we propose an efficient finite difference scheme for solving the cahn hilliard equation with a variable mobility in complex domains our method employs a type of unconditionally gradient stable splitting discretization we also extend the scheme to compute the cahn hilliard equation in arbitrarily shaped domains we prove the mass conservation property of the proposed discrete scheme for complex domains the resulting discretized equations are solved using a multigrid method numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly furthermore the multigrid efficiency is retained even if the embedded domain is present