immersed finite element methods for 4th order differential equations we propose three new finite element methods for solving boundary value problems of 4th order differential equations with discontinuous coefficients typical differential equations modeling the small transverse displacement of a beam and a thin plate formed by multiple uniform materials are considered one important feature of these finite element methods is that their meshes can be independent of the interface between different materials finite element spaces based on both the conforming and mixed formulations are presented numerical examples are given to illustrate capabilities of these methods

transformation coefficients of permutation groups the eigenfunction method is used to calculate the transformation coefficients m 1 2 m1 m2 from the yamanouchi basis of the permutation group sf1 f2 to the sf1 f2 sf1 sf2 irreducible basis a program in fortran is written and tables of the transformation coefficients for the permutation group sf up to f 6 are given possible applications of the transformation coefficients are sketched

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

split standard transformation coefficients the block selective conjecture the split basis for the symmetric group sn is adapted to sn and to the product subgroup sn a sa we consider matrices transforming between split bases and the standard young yamanouchi basis of the symmetric group sn we present a conjecture for directly relating those transformation matrices to a representation matrix in the young yamanouchi basis of a cycle permutation this block selective conjecture agrees with previously obtained solutions for the transformation matrices up to and including s6 in particular the conjecture provides a correct solution for the early occurrences of multiplicities in the product sn a sa to motivate the main result of this paper we first present transformations between permuted symmetric groups

the gel fand basis and matrix elements of the graded unitary group u m n projection operators with correct normalisation are constructed for obtaining the gel fand basis mod nu m of the graded unitary group u m n each u m n gel fand basis vector mod nu m for an f particle system uniquely corresponds to a non standard basis vector mod nu m of the permutation group s f the matrix element of the generator ei 1i of u m n between the two gel fand basis vectors mod nu m and mod nu m is proportional to the overlap between the two non standard basis vectors mod nu m and mod nu m of s f explicit formulae are given for the normalisation constant in the projection operator as well as for the matrix elements of the generator ei 1i of the graded unitary group u m n which is the extension of the gel fand tsetlin formula for the ordinary unitary group u m

super riemann surfaces uniformization and teichmuller theory teichmuller theory for super riemann surfaces is rigorously developed using the supermanifold theory of rogers in the case of trivial topology in the soul directions relevant for superstring applications the following results are proven the super teichmuller space is a complex super orbifold whose body is the ordinary teichmuller space of the associated riemann surfaces with spin structure for genusg 1 it has 3g 3 complex even and 2g 2 complex odd dimensions the super modular group which reduces super teichmuller space to super moduli space is the ordinary modular group there are no new discrete modular transformations in the odd directions the boundary of super teichmuller space contains not only super riemann surfaces with pinched bodies but rogers supermanifolds having nontrivial topology in the odd dimensions as well we also prove the uniformization theorem for super riemann surfaces and discuss their representation by discrete supergroups of fuchsian and schottky type and by beltrami differentials finally we present partial results for the more difficult problem of classifying super riemann surfaces of arbitrary topology

the numerical solutions of interface problems by infinite element method in this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems furthermore approximations of stress intensity factors are given the infinite element method may be considered as a certain scheme of mesh refinement but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently and that an approximate solution which has a singularity at the singular point can be also obtained

a general method to calculate all irreducible representations of a finite group a so called stochastic method is introduced which can be used to decompose the reducible representations of any finite group into irreducible representations in this way the regular representation of a finite group is reduced completely to obtain all of the irreducible representations of the group it also induces a new approach to calculate the character table of a finite group based on this method these calculations or decompositions can be performed by means of a computer

point group symmetrized boson representation algebraic solution for symmetry adapted bases of oh a point group symmetrized boson representation sbr is introduced that is particularly convenient for describing molecular vibrations in this paper the sbr is elucidated using the example of the molecule sf6 with oh symmetry the advantages of the sbr are that its basis vectors have a clear physical picture their number is very small equal to one eighth of the dimension of the reducible representation for oh and the irreducible bases for any concrete cases can be obtained trivially from those for the general case without any projection all the irreducible bases for the group chains oh d4 c4 or oh d4 d2 are tabulated once and for all as an application the hamiltonian in the algebraic model of iachello and oss for stretching vibrations of the molecule sf6 is diagonalized in the symmetry adapted bases

the immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources we extend the immersed interface method of leveque and li to find numerical solutions of one dimensional parabolic partial differential equations of the form u t beta x t u x x lambda x t u x kappa u u x f x where beta u beta u x and f may have known discontinuities at a known location x alpha at each time step a large weakly nonlinear system is set up using a difference scheme which is standard away from x alpha and which is derived for grid points near alpha by solving small linear systems which are determined from the jumps at x alpha the time stepping is done with a crank nicholson scheme and the nonlinear systems are solved with a levenberg marquardt method as an example we consider the flow of cars on a one lane highway with an entrance or exit where traffic is treated as a continuous fluid numerical examples show that we can compute solutions to these equations with second order accuracy

interaction of oscillating filaments a computational study abstract the immersed boundary technique is used to model the interaction of swimming filaments in a viscous incompressible fluid fluid quantities are represented on a grid and the immersed filaments are each represented by a discrete collection of moving points the effects of phase differences and proximity of the filaments on energy dissipation and swimming speeds are studied the computed results are compared with previous asymptotic analysis furthermore we present case studies which exhibit phase un locking phenomena undetectable using previous asymptotic analysis

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

theoretical and numerical analysis on a thermo elastic system with discontinuities a second order accurate numerical scheme is proposed for a thermo elastic system which models a bar made of two distinct materials the physical parameters involved may be discontinuous across the joint of the two materials where there might be also singular heat and or force sources the solution components the temperature and the displacement may change rapidly across the joint by transforming the system into a different one time marching schemes can be used for the new system which is well posed the immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources the proposed numerical method can fit both explicit and implicit formulation for the implicit version a stable and fast prediction correction scheme is also developed convergence analysis shows that our method is second order accurate at all grid points in spite of the discontinuities across the interface numerical experiments are performed to support the theoretical analysis in this paper

intrinsic lie group and nuclear collective rotation about intrinsic axes for any lie group g is so called intrinsic lie group g is introduced g and g being commutative and anti isomorphic it is shown that in the parameter space of a lie group g the corresponding intrinsic lie group g is just the second parameter group the general relations between the first and second parameter groups are derived in a simple way for the group so 3 there is the intrinsic lie group so 3 the infinitesimal generators of so 3 are precisely the components of the angular momentum in the intrinsic coordinate system therefore the intrinsic lie group so 3 provides an appropriate mathematical formalism for the description of collective rotations of nuclei about their intrinsic axes

coefficients of fractional parentage for u m p n q u m n u p q and u m n u m u n the outer product reduction coefficients orc which reduce the representation rep induced from the irreps of the permutation groups s f1 and s f2 into the irreps of s f1 f2 are shown to be the indirect coupling coefficients for the u m p n q u m n u p q irreducible basis the non standard orc for reducing the rep induced from the non standard irreps of s f13 s f1 s f3 and s f24 s f2 s f4 into that of s f s f12 s f34 with fij fi fj f f12 f34 are identified with the u f u f12 u f34 clebsch gordan coefficient for the special gel fand bases of u f12 and u f34 the u m p n q u m n u p q cfp as well as its special case the u m p u m u p cfp are identified with the s f s f12 s f34 outer product isoscalar factor the u m n u m u n cfp are obtained from the u m n u m u n cfp by simply changing all the partition labels for u n into their conjugates interchanging rows with columns and taking into account a phase change the cfp can be calculated from the orc numerical values of the one body cfp for systems with up to six particles are tabulated

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

error estimates in weighted sobolev norms for finite element immersed interface methods abstract when solving elliptic partial differential equations in a region containing immersed interfaces possibly evolving in time it is often desirable to approximate the problem using an independent background discretisation not aligned with the interface itself optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface at the cost of a higher complexity in the implementation a much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions as done in immersed boundary methods these methods are known to have inferior convergence properties depending on the global regularity of the solution across the interface when compared to enriched methods in this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon restricted to a small neighbourhood of the interface in particular we show that optimal approximations can be constructed in a natural and inexpensive way simply by reformulating the problem in a distributionally consistent way and by resorting to weighted norms when computing the global error of the approximation

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

multibody operator matrix elements and subduction coefficients in u n ii an approach to decomposing the specialized types of u n 3 j coefficients which arise in operator matrix element calculations in the gel fand basis is presented previous work on matrix elements of u n generator products will be extended to cases involving three generators and beyond in addition the methods of evaluation of su m su n su m n subduction coefficients will be applied the techniques are easily formulated in terms of yutsis graphs explicit phase relations are incorporated throughout the treatment

proof of polyakov conjecture on supercomplex plane using neumann series we solve iteratively the super beltrami equations sbe to arbitrary order then applying this we compute the energy momentum tensor and n points function for generic n starting from wzp action on the supercomplex plane we solve the superconformal ward identity and we show that the iterative solution to arbitrary order is resumed by wzp action this proves the polyakov conjecture on supercomplex plane

three dimensional immersed finite element methods for electric field simulation in composite materials this paper presents two immersed finite element ife methods for solving the elliptic interface problem arising from electric field simulation in composite materials the meshes used in these ife methods can be independent of the interface geometry and position therefore if desired a structured mesh such as a cartesian mesh can be used in an ife method to simulate 3 d electric field in a domain with non trivial interfaces separating different materials numerical examples are provided to demonstrate that the accuracies of these ife methods are comparable to the standard linear finite element method with unstructured body fit mesh copyright 2005 john wiley sons ltd

super beltrami differentials superconformal structures on a given 2 2 dim supermanifold give rise to the notion of super riemann surfaces srs s we investigate super beltrami coefficients which parametrize almost superconformal structures on the supermanifold the integrability condition of the structure reduces to a simple relation among the coefficients taking this into account we can write the super beltrami equations in a transparent form then an analysis of these equations enables us to prove the possibility of the special gauge choice of the wess zumino type for the super beltrami differentials this gauge choice simplifies the description of the deformations of srs s considerably and its existence will afford a better understanding of the structure of the super teichmuller space

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

the clebsch gordan coefficients and isoscalar factors of the graded unitary group su m n it is pointed out that the clebsch gordan coefficients cgc of the permutation group are the coefficients which couple irreducible bases irb of su m n and su p q to the irb of su mp nq mp np and the outer product reduction coefficients orc of the permutation group are the coefficients which couple the irb of su m n and su p q into the irb of su m p n q the orc of permutation groups with due modification in signs are the cgc for the special gel fand basis of su m n the isoscalar factor isf for the permutation group chain s f contains implies s f1 s f2 is the isf for the graded unitary group chain su mp nq mq np contains implies su m n su p q and the outer product isf for the group chain s f contains implies s f1 s f2 is the isf for su mp nq mq np contains implies su m n x su p q all these coefficients can be calculated easily and some are already available

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

su mn su m su n isoscalar factors and s f1 f2 s f1 s f2 isoscalar factors a simple relation is found between the isoscalar factor isf of the unitary group and those of the permutation group i e the su mn su m su n isf is equal to the s f1 f2 s f1 s f2 isf since the values of s f1 f2 s f1 s f2 isf are independent of m and n one arrives at an important conclusion that the values of su mn su m su n isf are also independent of m and n therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time an eigenfunction metho for evaluating the su mn su m su n isf is given which can be easily translated into a computer program

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a petrov galerkin finite element method for interface problems arising in sensitivity computations continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design to fast algorithms in computational fluid dynamics to the quantification of uncertainty in order to make use of these methods for interface problems one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces in this paper we develop a discontinuous petrov galerkin finite element scheme for solving the sensitivity equation resulting from a 1d interface problem the 1d example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities in particular the sensitivity boundary value problem must be formulated in a very weak sense and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes numerical examples are presented to illustrate the benefits of this approach

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

linear and bilinear immersed finite elements for planar elasticity interface problems this article is to discuss the linear which was proposed in 18 19 and bilinear immersed finite element ife methods for solving planar elasticity interface problems with structured cartesian meshes basic features of linear and bilinear ife functions including the unisolvent property will be discussed while both methods have comparable accuracy the bilinear ife method requires less time for assembling its algebraic system our analysis further indicates that the bilinear ife functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear ife functions numerical examples are provided to demonstrate that both linear and bilinear ife spaces have the optimal approximation capability and that numerical solutions produced by a galerkin method with these ife functions for elasticity interface problem also converge optimally in both l 2 and semi h 1 norms

the casimir invariants and gel fand basis of the graded unitary group su m n in the space of f particle product states the casimir invariants of the graded unitary group su m n are functions of the class operators of the graded coordinate permutation group s f which are equal to the class operators of the graded state permutation group s f formulae for the quadratic and cubic casimir invariants are given it is shown that in general the eigenvalue formulae for the u m n casimir invariants as functions of the partition can be obtained from those for u m n by simply changing the argument of the functions from m n to m n the quasi standard basis of s f is identified with the gel fand basis of su m n i e the irreducible basis classified according to the group chain su m n contains implies su m n 1 contains implies su m contains implies su 2 contains implies u 1 the special gel fand basis of su m n for which the weight is restricted to 1 1 1 can be easily obtained from the gel fand basis of su m n with due allowances for modifying sign factors arising from the grading the construction of a general su m n gel fand basis is also discussed

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

the clebsch gordan coefficients of permutation groups s 2 s 6 the clebsch gordan cg coefficients of the permutation groups s 2 s 6 are calculated by using the eigenfunction method phases have been chosen in a consistent way to exhibit as many symmetries for the coefficients as is possible including non multiplicity free cases the cg coefficients of s 2 s 5 and those for 321 321 to 321 5 of s 6 are tabulated in the square root form of rationals these tables together with the s 6 contains implies s 5 isoscalar factors published earlier provide a complete tabulation of the s 2 s 6 cg coefficients given in the young yamanouchi basis

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

a new approach to permutation group representations ii the formula for the yamanouchi matrix elements is rederived by the eigenfunction method in a simple fashion

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

polyakov conjecture on the supertorus we prove the polyakov conjecture on the supertorus st2 we determine an iterative solution at any order of the superconformal ward identity and we show that the polyakov action that describes the 1 0 2d two dimensional supergravity resums this perturbative series the resolution of the super beltrami equation for the wess zumino field from which the polyakov action is expressed is done by using on the one hand the cauchy kernel techniques on st2 defined in h kachkachi and m kachkachi class quantum grav 11 493 1994 and on the other hand the formalism developed in m kachkachi and s kouadik j math phys 38 4336 1997 hence we determine the n points green functions from the polyakov action expressed as a functional integral of the beltrami superfield

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

gradient recovery for elliptic interface problem ii immersed finite element methods abstract this is the second paper on the study of gradient recovery for elliptic interface problem in our previous work guo and yang 2016 17 we developed a novel gradient recovery technique for finite element method based on the body fitted mesh in this paper we propose new gradient recovery methods for two immersed interface finite element methods symmetric and consistent immersed finite method ji et al 2014 23 and petrov galerkin immersed finite element method hou et al 2004 22 and hou and liu 2005 20 compared to the body fitted mesh based gradient recovery method the new methods provide a uniform way of recovering gradient on regular meshes numerical examples are presented to confirm the superconvergence of both gradient recovery methods moreover they provide asymptotically exact a posteriori error estimators for both immersed finite element methods

a cartesian grid nonconforming immersed finite element method for planar elasticity interface problems in this paper a new nonconforming immersed finite element ife method on triangular cartesian meshes is developed for solving planar elasticity interface problems the proposed ife method possesses optimal approximation property for both compressible and nearly incompressible problems its degree of freedom is much less than those of existing finite element methods for the same problem moreover the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh both theory and numerical experiments are presented to demonstrate the effectiveness of the new method theoretically the unisolvent property and the consistency of the ife space are proved experimentally extensive numerical examples are given to show that the approximation orders in l2 norm and semi h1 norm are optimal under various lam parameters settings and different interface geometry configurations

a conformal decomposition finite element method for arbitrary discontinuities on moving interfaces summary interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities the conformal decomposition fem decomposes multimaterial elements of a non conforming background mesh into sub elements that conform to material interfaces captured using a level set method as the interface evolves interfacial nodes move and background nodes may change material the present work describes approaches for handling moving interfaces in the context of the conformal decomposition fem for both weakly and strongly discontinuous fields dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first order and second order time integration methods the moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one dimensional and two dimensional test problems while achieving the expected second order error convergence rate in space and time copyright 2014 john wiley sons ltd

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres