finite element approximation of a phase field model for void electromigration we consider a fully practical finite element approximation of the nonlinear degenerate parabolic system begin align gamma textstyle frac partial u partial t nabla b u nabla w alpha phi 0 quad w gamma delta u gamma 1 psi u nonumber quad nabla c u nabla phi 0 nonumber end align subject to an initial condition u 0 cdot in 1 1 on u and flux boundary conditions on all three equations here gamma in mathbb r 0 alpha in mathbb r geq 0 psi is a nonsmooth double well potential and c u 1 u b u 1 u 2 are degenerate coefficients diffusional mobility the degeneracy in b restricts u cdot cdot in 1 1 the above in the limit gamma rightarrow 0 models the evolution of voids by surface diffusion in an electrically conducting solid in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenera

characteristic cut finite element methods for convection diffusion problems on time dependent surfaces we develop a finite element method for convection diffusion problems on a given time dependent surface for instance modeling the evolution of a surfactant the method is based on a characteristic

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a fully discrete evolving surface finite element method in this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface in earlier papers using a suitable variational formulation in time dependent sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces ima j numer anal 25 2007 pp 385 407 math comp to appear optimal order l 2 gamma t and h 1 gamma t error bounds were proved for linear finite elements in this work we prove optimal order error bounds for a backward euler time discretization

a convergent finite volume scheme for diffusion on evolving surfaces a finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed the underlying motion is assumed to be described by a fixed not necessarily normal velocity field the ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes where grid vertices move on motion trajectories a consistent finite volume discretization of the induced transport on the simplices and a proper incorporation of a diffusive flux balance at simplicial faces the semi implicit scheme is derived via a discretization of the underlying conservation law and discrete counterparts of continuous a priori estimates in this geometric setup are proved the continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes furthermore numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results finally we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications

transport and diffusion of material quantities on propagating interfaces via level set methods we develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework level set methods are computational techniques for tracking moving interfaces they work by embedding the propagating interface as the zero level set of a higher dimensional function and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes the traditional level set method works in the trace space of the evolving interface and hence disregards any parameterization in the interface description consequently material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework we develop model equations and algorithmic techniques to extend the level set method to include these problems we demonstrate the accuracy of our approach through a series of test examples and convergence studies

dead beat control of linear discrete time systems the paper considers the problem of determining the various classes of state feed back dead beat controllers for linear multivariable discrete time systems unlike earlier work attention is not restricted to the special case of minimum time dead beat controllers a general dead beat control problem is posed and solved the solution is based principally on an approach of closed loop eigenstructure assignment the minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix the concept of the greatest minimum of free parameters is introduced finally the method is illustrated by a numerical example

delay transition of a recurrent bus on a circular route abstract we study the behavior of a recurrent bus on a circular route with many bus stops we present the basic model determining the recurrent time one period on each bus stop the recurrent times on m bus stops are given by m simultaneous nonlinear equations it is found that the dynamical phase transition between the delay and schedule time phases occurs with varying the initial period the dynamical transition depends on number m of bus stops when m is larger than the critical value the recurrent time diverges with the number of rotation the recurrent time is described approximately in terms of a nonlinear map the schedule time phase is determined by the basin of attraction

dynamical transition to periodic motions of a recurrent bus induced by nonstops abstract we study the dynamical behavior of a recurrent bus on a circular route with many bus stops when the recurrent bus passes some bus stops without stopping the recurrent time one period is described in terms of a nonlinear map it is shown that the recurrent bus exhibits the complex periodic behaviors the dynamical transitions to periodic motions occur by increasing nonstops the periodic motions depend on the property of an attractor of the nonlinear map the period n of the attractor varies sensitively with the number of nonstops

transition to chaotic motion of a cyclic bus induced by nonstops abstract we study the dynamical behavior of a cyclic bus on a circular route with many bus stops when the cyclic bus passes some bus stops without stopping to retrieve the delay the recurrent time one period is described in terms of a piecewise nonlinear map it is found that the cyclic bus exhibits a dynamical transition to chaotic behavior when the number of bus stops without stopping is larger than the critical value it is shown that there are three distinct dynamical states the schedule time phase convergence of the recurrent time the delay phase divergence of the recurrent time and the chaotic motion liapunov exponent is calculated for distinct dynamical states the chaotic motion depends on the property of an attractor of the nonlinear map

analysis of a continuous car following model for a bus route existence stability and bifurcations of synchronous motions abstract this paper studies the synchronous motions motions for which each of the buses moves with a constant velocity of a car following model for a circular bus route different types of synchronous motion are identified and their local stability is studied using root locus methods the bifurcations of synchronous motions are described and simulation results are given

parametric time optimal control of linear discrete time systems by state feedback part 2 irregular kronecker invariants in an earlier paper part 1 a parametric state feedback time optimal control law was devised for the regular kronecker invariants on the basis of parametrization of the vector companion form the present paper describes a straight forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular kronecker invariants hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law an illustrative example is presented

error analysis of a space time finite element method for solving pdes on evolving surfaces in this paper we present an error analysis of an eulerian finite element method for solving parabolic partial differential equations pdes posed on evolving hypersurfaces in rd d 2 3 the method employs discontinuous piecewise linear in time continuous piecewise linear in space finite elements and is based on a space time weak formulation of a surface pde problem trial and test surface finite element spaces consist of traces of standard volumetric elements on a space time manifold resulting from the evolution of a surface we prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm furthermore we derive regularity results for solutions of parabolic pdes on an evolving surface which we need in a duality argument used in the proof of the second order convergence estimate

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

an eulerian space time finite element method for diffusion problems on evolving surfaces in this paper we study numerical methods for the solution of partial differential equations on evolving surfaces the evolving hypersurface in bbb r d defines a d dimensional space time manifold in the space time continuum bbb r d 1 we derive and analyze a variational formulation for a class of diffusion problems on the space time manifold for this variational formulation new well posedness and stability results are derived the analysis is based on an inf sup condition and involves some natural but nonstandard anisotropic function spaces based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous galerkin dg space time finite element discretization this dg space time method is explained and results of numerical experiments are presented that illustrate its properties

the effect of sampling period on the behaviour of systems incorporating state feedback an exposition of the effects of sampling on the controllability characteristics and the kronecker invariants of the plant and the input matrices of sampled data systems is presented in particular the effect of sampling period on the feedback matrix for time optimal control of sampled data systems is investigated in detail and it is shown that different classes of systems produce different response characteristics as the sampling period becomes large also simple expressions for a nt in terms of a t and b nt as a function of a t and b t are derived these expressions are computationally attractive and therefore facilitate the study of the effect of sampling period on system behaviour by applying these expressions the effects of sampling period on the feedback matrices for time optimal control of three classes of systems whose open loop plants are asymptotically stable marginally stable and unstable are illustrated by numerical examples

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

unfitted finite element methods using bulk meshes for surface partial differential equations in this paper we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements the key idea is that the n dimensional hypersurface gamma subset mathbb r n 1 is embedded in a polyhedral domain in mathbb r n 1 consisting of a union mathscr t h of n 1 simplices the unifying feature of the methodological approach is that the finite element approximating space is based on continuous piecewise linear finite element functions on the bulk triangulation mathscr t h which is independent of gamma our first method is a sharp interface method sif which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation gamma h of gamma the full gradient is used rather than the projected tangential gradient and it is this which distinguishes sif from the method of m a olshanskii a reusken and j grande siam j nu

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

stability and error analysis for a diffuse interface approach to an advection diffusion equation on a moving surface in this paper we analyze a fully discrete numerical scheme for solving a parabolic pde on a moving surface the method is based on a diffuse interface approach that involves a level set description of the moving surface under suitable conditions on the spatial grid size the time step and the interface width we obtain stability and error bounds with respect to natural norms furthermore we present test calculations that confirm our analysis