a new fast physical optics for smooth surfaces by means of a numerical theory of diffraction a new technique to compute the physical optics po integral is presented the technique consists of a blind code that computes the different contributions stationary phase points end points etc numerically this technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane this algorithm permits a fast and accurate evaluation of the po integral for smooth large surfaces the cpu time is almost independent of frequency

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

viscous cahn hilliard equation ii analysis the viscous cahn hilliard equation may be viewed as a singular limit of the phase field equations for phase transitions it contains both the allen cahn and cahn hilliard models of phase separation as particular cases by specific choices of parameters it may be formulated as a one parameter say homotopy connecting the cahn hilliard 0 and allen cahn 1 models the limit 0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators the properties of the gradient dynamical system generated by the viscous cahn hilliard equation are studied as varies in 0 1 continuity of the phase portraits near equilibria is established independently of 0 1 and using this a piecewise uniform in time perturbation result is proved for trajectories finally the continuity of the attractor is established and in one dimension the existence and continuity of inertial manifolds shown and the flow on the attractor detailed

isogeometric analysis of the cahn hilliard equation a convergence study herein we present a numerical convergence study of the cahn hilliard phase field model within an isogeometric finite element analysis framework using a manufactured solution a mixed formulation of the cahn hilliard equation and the direct discretisation of the weak form which requires a c 1 continuous approximation are compared in terms of convergence rates for approximations that are higher than second order in space the direct discretisation is found to be superior suboptimal convergence rates occur when splines of order p 2 are used this is validated with a priori error estimates for linear problems the convergence analysis is completed with an investigation of the temporal discretisation second order accuracy is found for the generalised method this ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the cahn hilliard equation the isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition

edge diffraction in nurbs utd method the edge diffraction in non uniform rational b spline uniform geometrical theory of diffraction nurbs utd method is studied in this communication basic aspects like ray tracing shadowing test and calculating of ray field are managed through numerical measures respectively in order to solve the scattering problem of electrically large objects compared to utd method where objects are modeled by the combination of boards cylinders and cones the use of nurbs provides accurate modeling of objects and can enable nurbs utd method to be directly applied to arbitrary electrically large convex surfaces the computational accuracy can also be improved and numerical results show the validation and the improvement

an efficient algorithm for the computation of the utd t transition function we present an efficient algorithm for the numerical calculation of the t canonical transition function which is encountered in the uniform geometrical theory of diffraction utd such a transition function allows the uniform ray field description of various high frequency diffraction mechanisms such as double wedge or vertex diffraction the proposed algorithm is valid for both real and imaginary arguments as required to deal with the general case in utd applications

representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples it is shown that the electromagnetic em field radiated or scattered by bounded sources can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded the number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field this result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain it is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry the case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered for these geometries precise and fast sampling algorithms of central type are presented which allow an efficient recovery of em fields from a nonredundant finite number of samples such algorithms are stable with respect to random errors affecting the data

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

algorithm for the computation of the generalized fresnel integral the generalized fresnel integral appears as a canonical function in the uniform ray field representation of several high frequency diffraction mechanisms such as double wedge or vertex diffraction here we propose an algorithm for its calculation which is valid both for real and imaginary arguments as required for treating the general case in the uniform geometrical theory of diffraction framework

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

combining the moment method with geometrical modelling by nurbs surfaces and bezier patches induced current distributions on conducting bodies of arbitrary shape modelled by nurbs non uniform rational b splines surfaces are obtained by using a moment method approach to solve an electric field integral equation efie the nurbs surfaces are expanded in terms of bezier patches by applying the cox de boor transformation algorithm this transformation is justified because bezier patches are numerically more stable than nurbs surfaces new basis functions have been developed which extend over pairs of bezier patches these basis functions can be considered as a generalization of rooftop functions the method is applied to obtain rcs values of several objects modelled with nurbs surfaces good agreement with results from other methods is observed the method is efficient and versatile because it uses geometrical modelling tools that are quite powerful

analysis of antennas on board arbitrary structures modeled by nurbs surfaces an accurate and efficient numerical scheme has been developed for predicting high frequency radiation patterns of antennas mounted on arbitrary structures modeled by parametric surfaces the method is based on geometric optics go and the uniform theory of diffraction utd nonuniform rational b splines nurbs surfaces have been used to describe the geometry of the structure as most of the computer aided geometric design cagd tools available in the industry are biased on nurbs the scheme can perform the electromagnetic analysis without any new or additional remeshing of the geometrical model a special ray tracing technique that combines go and utd with nurbs has been developed this technique uses some selective criteria in order to identify rapidly the nurbs where a ray impact may occur impact point coordinates are obtained by means of an optimization procedure based on the conjugate gradient method cgm the accuracy and efficiency of the approach are shown comparing it with other methods

combining the higher order method of moments with geometric modeling by nurbs surfaces this paper introduces the nonuniform rational b spline nurbs surfaces to improve the geometric modeling of the higher order method of moments mom the electric field integral equation efie is discretized by the hierarchical higher order basis functions and converted to a matrix equation then the elements of the impedance matrix are efficiently evaluated by a new set of formulas the bistatic radar cross sections rcs obtained by this new technique are compared with those obtained by the commonly used higher order mom the example of a cylinder and a missile shows excellent accuracy of the nurbs surfaces and that of the resultant rcs moreover this new technique can fully exploit the flexibility of the higher order basis functions when the surface is highly curved whereas the commonly used higher order mom can not

inexact hierarchical scale separation a two scale approach for linear systems from discontinuous galerkin discretizations abstract hierarchical scale separation hss is an iterative two scale approximation method for large sparse systems of linear equations arising from discontinuous galerkin dg discretizations hss splits the linear system into a coarse scale system of reduced size corresponding to the local mean values of the solution and a set of decoupled local fine scale systems corresponding to the higher order solution components this scheme then alternates between coarse scale and fine scale system solves until both components converge the motivation of hss is to promote parallelism by decoupling the fine scale systems and to reduce the communication overhead from classical linear solvers by only applying them to the coarse scale system we propose a modified hss scheme inexact hss ihss that exploits the highly parallel fine scale solver more extensively and only approximates the coarse scale solution in every iteration thus resulting in a significant speedup the tolerance of the coarse scale solver is adapted in every ihss cycle controlled by the residual norm of the fine scale system anderson acceleration is employed in the repeated solving of the fine scale system to stabilize the scheme we investigate the applicability of ihss to systems stemming from the nonsymmetric interior penalty dg discretization of the cahn hilliard equation discuss its hybrid parallel implementation for large scale simulations and compare the performance of a widely used iterative solver with and without ihss

uniform asymptotic evaluation of surface integrals with polygonal integration domains in terms of utd transition functions the field scattered by a scattering body or by an aperture in the free space or in an unbounded homogenous medium can be described in terms of a double integral in this paper we show how a canonical integral on a polygonal domain with a constant amplitude function and a quadratic phase variation can be exactly expressed in terms of special functions namely fresnel integrals and generalized fresnel integrals this exact reduction represents a paradigm for deriving a new asymptotic evaluation for a more general integral this new asymptotic uniform integral evaluation is expressed in the format of the uniform geometrical theory of diffraction which is convenient for numerical computations

hp version interior penalty dgfems for the biharmonic equation we construct hp version interior penalty discontinuous galerkin finite element methods dgfems for the biharmonic equation including symmetric and nonsymmetric interior penalty discontinuous galerkin methods and their combinations semisymmetric methods our main concern is to establish the stability and to develop the a priori error analysis of these methods we establish error bounds that are optimal in h and slightly suboptimal in p the theoretical results are confirmed by numerical experiments

analysis of antenna around nurbs surface with hybrid mom po technique the hybrid method of moments and physical optics mom po approach is used to calculate the radiation pattern of antenna around arbitrarily shaped structure the structure is modeled with non uniform rational b spline nurbs surfaces the hybrid mom po approach is implemented by modifying the impedance matrix of the mom region with po formula for the scattered po field is deduced for cases of antenna located around nurbs surface the stationary phase method spm is applied for the integral of the induced current in the po region results obtained from this method and from mom po approach based on triangle facet model agree well while the former is more efficient in execution time

shape reconstruction of three dimensional conducting curved plates using physical optics nurbs modeling and genetic algorithm a microwave inverse scattering problem including a method for shape reconstruction of three dimensional electrically large conducting patches with simple geometries using genetic algorithm is presented unknown shape reconstruction algorithm starts from the knowledge of the simulated radar cross section rcs data through back scattering far field computation using physical optics approximation the forward problem involves the computation of the multiple frequency and multiple direction rcs of three dimensional large conducting patches modeled by nonuniform rational b spline nurbs surfaces the control points of nurbs are the geometrical parameters which are optimized for the shape reconstruction procedure the extended stationary phase method and critical cases which occur in physical optics computations in the forward problem are also discussed noise effect and the influence of increment in the number of control points of a nurbs over the inversion algorithm are investigated as well numerical results are presented to verify the operation of the proposed algorithm

efficient algorithm for the evaluation of the physical optics scattering by nurbs surfaces with relatively general boundary condition an adaptive integration algorithm is presented for the computation of the physical optics po electric and magnetic field scattered by electrically large objects modeled by non uniform rational b splines nurbs the algorithm is the customization of a more general purpose result that has been recently published by using a unique formulation both impenetrable e g impedance surfaces coated conductors as well as transparent thin sheet materials e g thin dielectric panels or frequency selective surfaces are treated via their fresnel reflection and transmission coefficients the po radiation integral is evaluated over the nurbs parametric domain since most of the computer aided geometric design cagd tools are based on nurbs the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description with no need of generating new geometrical models furthermore the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard nyquist based sampling integration algorithms such reduction of the sampling points is achieved by resorting to high frequency technique concepts and allows a significant reduction of the cpu computational burden therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real life electrically large objects

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

form factors for quasi particles in c 1 conformal field theory the non fermi liquid physics at the edge of fractional quantum hall systems is described by specific chiral conformal field theories with central charge c 1 the charged quasi particles in these theories have fractional charge and obey a form of fractional statistics in this paper we study form factors which are matrix elements of physical conformal operators evaluated in a quasi particle basis that is organized according to the rules of fractional exclusion statistics using the systematics of jack polynomials we derive selection rules for a special class of form factors we argue that finite temperature green functions can be evaluated via systematic form factor expansions using form factors such as those computed in this paper and thermodynamic distribution functions for fractional exclusion statistics we present a specific case study where we demonstrate that the form factor expansion shows a rapid convergence

a form factor approach to finite temperature correlation functions in c 1 cft the excitation spectrum of specific conformal field theories cft with central charge c 1 can be described in terms of quasi particles with charges q p 1 and fractional statistics properties using the language of jack polynomials we compute form factors of the charge density operator in these cfts we study a form factor expansion for the finite temperature density density correlation function and find that it shows a quick convergence to the exact result the low temperature behaviour is recovered from a form factor with p 1 particles while the high temperature limit is recovered from states containing no more than three particles

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

on new triangle quadrature rules for the locally corrected nystrom method formulated on nurbs generated bezier surfaces in 3 d nonuniform rational b splines nurbs are the most widely used technique in today s geometric computer aided design systems for modeling surfaces combining the locally corrected nystrom lcn method with nurbs requires formulating lcn on both quadrilateral and triangular bezier surfaces as a typical nurbs generated bezier mesh includes elements of both the types while on quadrilateral elements the product of 1 d gaussian quadrature rules can be applied to lcn effectively gaussian integration rules available for triangles cannot efficiently be applied to lcn for two reasons first they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order second they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles in this paper we construct a new set of quadrature rules for bezier triangles i e degenerate quadrilaterals based on the newton cotes equidistant quadrature rules and apply these rules to the lcn solution of the electric magnetic and combined field integral equations results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to lcn effectively for orders from 0 to 9 inclusively

error analysis of a mixed finite element method for a cahn hilliard hele shaw system we present and analyze a mixed finite element numerical scheme for the cahn hilliard hele shaw equation a modified cahn hilliard equation coupled with the darcy flow law this numerical scheme was first reported in feng and wise siam j numer anal 50 1320 1343 2012 with the weak convergence to a weak solution proven in this article we provide an optimal rate error analysis a convex splitting approach is taken in the temporal discretization which in turn leads to the unique solvability and unconditional energy stability instead of the more standard ell infty 0 t l 2 cap ell 2 0 t h 2 l 0 t l2 l2 0 t h2 error estimate we perform a discrete ell infty 0 t h 1 cap ell 2 0 t h 3 l 0 t h1 l2 0 t h3 error estimate for the phase variable through an l 2 l2 inner product with the numerical error function associated with the chemical potential as a result an unconditional convergence for the time step tau in terms of the spatial resolution h is derived the nonlinear analysis is accomplished with the help of a discrete gagliardo nirenberg type inequality in the finite element space gotten by introducing a discrete laplacian delta h h of the numerical solution such that delta h phi in s h h sh for every phi in s h sh where s h sh is the finite element space

a hybrid high order method for the cahn hilliard problem in mixed form in this work we develop a fully implicit hybrid high order algorithm for the cahn hilliard problem in mixed form the space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces the proposed method has several advantageous features i it supports fairly general meshes possibly containing polyhedral elements and nonmatching interfaces ii it allows arbitrary approximation orders and iii it has a moderate computational cost thanks to the possibility of locally eliminating element based unknowns by static condensation we perform a detailed stability and convergence study proving optimal convergence rates in energy like norms numerical validation is also provided using some of the most common tests in the literature

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

on the quasiparticle description of c 1 cfts we show that the description of c 1 conformal field theory in terms of quasiparticles satisfying fractional statistics can be obtained from the sine gordon model with a chemical potential a in the limit where a m these quasiparticles are related to the excitations of the calogero sutherland cs model we provide a direct calculation of their two particle s matrix using korepin s method we also reconsider the computation of the cs s matrix in terms of particles with fractional charge

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

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

application of the characteristic basis function method utilizing a class of basis and testing functions defined on nurbs patches a novel implementation of the characteristic basis function method cbfm is given in which the high level basis functions called characteristic basis functions cbfs are represented in terms of curved rooftops generated from nonuniform rational b splines nurbs in the parametric u v domain the associated macro testing functions are defined by using curved razor blade functions corresponding to each rooftop the underlying objective of the cbfm is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments mom the result is therefore an approach which can handle many complex cases via direct solvers without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill conditioned matrices as a result of the combination of the cbfm with the special class of low level basis and testing functions directly located over nurbs surfaces complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the cpu time as well as required memory resources

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

fully discrete dynamic mesh discontinuous galerkin methods for the cahn hilliard equation of phase transition fully discrete discontinuous galerkin methods with variable meshes in time are developed for the fourth order cahn hilliard equation arising from phase transition in materials science the methods are formulated and analyzed in both two and three dimensions and are proved to give optimal order error bounds this coupled with the flexibility of the methods demonstrates that the proposed discontinuous galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and non conforming plate finite element methods for solving fourth order partial differential equations

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

application of physical optics to the rcs computation of bodies modeled with nurbs surfaces the paper presents a method for the computation of the monostatic radar cross section rcs of electrically large conducting objects modeled by nonuniform rational b spline nurbs surfaces using the physical optic po technique the nurbs surfaces are expanded in terms of rational bezier patches by applying the cox de boor transform algorithm this transformation is justified because bezier patches are numerically more stable than nurbs surfaces the po integral is evaluated over the parametric space of the bezier surfaces using asymptotic integration the scattering field contribution of each bezier patch is expressed in terms of its geometric parameters excellent agreement with po predictions is obtained the method is quite efficient because it makes use of a small number of patches to model complex bodies so it requires very little memory and computing time

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

an improved error bound for a finite element approximation of a model for phase separation of a multi component alloy with a concentration dependent mobility matrix using a slightly different discretization scheme in time and adapting the approach in nochetto et al 1998 for analysing the time discretization error in the backward euler method we improve on the error bounds derived in i barrett and blowley 1998 and ii barrett and blowey 1999c for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi component alloy with a concentration dependent mobility matrix and i a logarithmic free energy and ii a non smooth free energy the deep quench limit respectively moreover the improved error bound in the deep quench limit is optimal numerical experiments with three components illustrating the above error bounds are also presented

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flows in this paper we further investigate the use of a fully discontinuous finite element discrete formulation for the study of shallow water free surface flows in the fully nonlinear and weakly dis persive flow regime we consider a decoupling strategy in which we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non hydrostatic effects this source term can be computed through the resolution of elliptic second order linear sub problems which only involve second order partial derivatives in space we then introduce an associated symmetric weighted internal penalty discrete bilinear form allowing to deal with the discontinuous nature of the elliptic problem s coefficients in a stable and consistant way similar discrete formulations are also introduced for several recent optimized fully nonlinear and weakly dis persive models these formulations are validated again several benchmarks involving h convergence p convergence and comparisons with experimental data showing optimal convergence properties

fully coupled hybrid fem utd method using nurbs for the analysis of radiation problems a novel hybrid finite element method fem uniform theory of diffraction utd method for the analysis of radiating structures in the presence of electrically large objects is presented the hybridization is done in a fully coupled way taking into account mutual interactions between the fem and utd regions the utd objects are modeled using non uniform rational b spline nurbs surfaces ray acceleration techniques and simple clustering strategies are used to speed up the computation time numerical results are presented showing the validity of the method

evaluation of physical optics integrals from b spline surfaces by means of a fast locating algorithm of stationary points a novel modeling method based on the parametric domain decomposition is proposed for the b spline surfaces this method combines the advantages of conventional b spline and bezier surfaces moreover a fast algorithm based on the new modeling method is presented to locate the stationary points including inner and boundary stationary points of physical optics po integrals from targets modeled with double quadratic b spline surfaces in this algorithm the locating problem is reduced to a numerical or analytical solution no approximation is involved in the proposed modeling method and locating algorithm the stationary phase method by means of the fast locating algorithm is employed to evaluate po integrals the cpu time is independent of the incident frequency and physical dimension of scattered surfaces the accuracy and performance of the presented methods are validated by comparing them with those of the brute force method

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

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

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

a diffuse interface model for axisymmetric immiscible two phase flow a diffuse interface model is considered for solving axisymmetric immiscible two phase flow with surface tension the navier stokes ns equations are modified by the addition of a continuum forcing the interface between the two fluids is considered as the half level set of a mass concentration c which is governed by the cahn hilliard ch equation a fourth order degenerate nonlinear parabolic diffusion equation in this work we develop a nonlinear multigrid method to solve the ch equation with degenerate mobility and couple this to a projection method for the incompressible ns equations the diffuse interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc cut and connect or other artificial procedures we present results for rayleigh s capillary instability up to forming satellite drops the results agree well with the linear stability theory

calculation of physical optics integrals over nurbs surface using a delaminating quadrature method physical optics po is a widely used radar cross section rcs estimation approach for electrically large scaled objects and it is finally reduced to the calculation of highly oscillatory integrals in this paper the po integrals over non uniform rational b spline nurbs surfaces is proposed to be calculated with a delaminating quadrature method the method has the merits of being very stable accurate and fast and its computation cost does not increase with frequency moreover in order to keep these good properties for po integrals involving critical points stationary phase point resonant point a scheme of integral subdividing is proposed and new approaches for locating the critical points are also presented numerical examples of flat surface cylindrical surface and spherical surface well show the advantages of the new method

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

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

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods