a characterization of ideal polyhedra in hyperbolic 3 space the goals of this paper are to provide a characterization of dihedral angles of convex ideal those with all vertices on the sphere at infinity polyhedra in h3 and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of h3 these characterizations are given in respectively theorems 0 1 and 10 5 the first theorem is proved in detail while the proof of the second which is similar is only outlined the results of this paper grow out of the general framework of the author s doctoral dissertation 13 as published in 20 a lot of the language and some of the auxilary results come from there as well so familiarity with the latter reference is very helpful the necessity of the conditions postulated in theorem 0 1 has been shown in 14 hence only the sufficiency direction will be shown in the current paper in 1832 jakob steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere this was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type by steinitz 24 however theorem 0 1 gives such a characterization section 11 is dedicated to a very brief historical survey and some purely graph theoretic and computational geometric consequences of theorem 0 1 in order to state this theorem suppose that a convex ideal polyhedron p in h3 is given let p denote the poincar dual of p and assign to each edge e of p a weight w e equal to the exterior dihedral angle at the corresponding edge e of p then the following result holds

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

surface integral equation method of moments with multiregion basis functions applied to plasmonics a surface integral equation method of moments sie mom approach is presented for the solution of multiple homogeneous penetrable bodies the problem is formulated using multiregion mr piecewise vector basis functions defined uniformly on the boundary surfaces interfaces and line junctions the mr bases are oriented functions implicitly satisfying the boundary conditions their use greatly simplifies the formulation and thereby the implementation for the solution of arbitrary complex composite problems having multiple junctions moreover the generation of the mr functions involves a negligible overhead at the setup numerical examples are presented to demonstrate the accuracy and versatility of sie mom with mr basis functions in plasmonics where this approach remains much unexplored

the locally stabilized finite volume method for incompressible flow a locally stabilized finite volume method for the two dimensional incompressible flow governed by stationary stokes and navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem the optimal l 2 error estimate of u h p h determined by stokes equations is also deduced some numerical examples are provided to check the efficiency of this method

on the teichmuller theory of circle patterns given a circle pattern on the riemann sphere in this paper we prove that its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

convergence analysis for the full upwind finite volume solution of a convection diffusion problem abstract the paper is devoted to the study of convergence properties for an often used cell centered full upwind finite volume method fvm with voronoi boxes this fvm is applied to a convection diffusion problem the approach to proving convergence of the fvm is based on the construction of a nonconforming petrov galerkin finite element method fem such that the system of linear equations coincides completely with that of the fvm thus by proving convergence properties of the fem we obtain similar ones for the fvm for the error estimation of the fem the second strang lemma has to be modified

nested equivalent source approximation for the modeling of penetrable bodies a multiregion nested equivalent source approximation nesa is presented for the fast analysis of electromagnetic scattering by piecewise homogeneous media nesa was introduced for the analysis of metallic problems at low and moderate frequencies and later extended to the fully dynamic case with the introduction of a directional scheme in this communication we modify the wideband scheme for the analysis of arbitrary penetrable media the asymptotic computational costs of the algorithm are compatible with the state of the art factorizations namely linear and quasi linear o n and o n log n for moderate and high frequency problems respectively numerical examples validate the accuracy and favorable computational cost of nesa when applied to penetrable bodies

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

the finite volume method based on stabilized finite element for the stationary navier stokes problem a finite volume method based on stabilized finite element for the two dimensional stationary navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem we obtain the well posedness of the fvm based on stabilized finite element for the stationary navier stokes equations moreover for quadrilateral and triangular partition the optimal h 1 error estimate of the finite volume solution u h and l 2 error estimate for p h are introduced finally we provide a numerical example to confirm the efficiency of the fvm

a comparative study of surface integral equations for accurate and efficient analysis of plasmonic structures surface integral equations which are commonly used in electromagnetic simulations have recently been applied to various plasmonic problems while there is still no complete agreement on which formulations provide accurate and efficient solutions in this paper we present the strong material dependences of the conventional formulations revealing their contradictory performances for different problems we further explain the numerical problems in the constructed matrix equations shedding light on the design of alternative formulations that can be more accurate efficient and stable than the existing ones based on our observations in the limit cases we present a new formulation namely a modified combined tangential formulation mctf which provides stable solutions of plasmonic problems in wide ranges of negative permittivity values the favorable properties of mctf in comparison to other formulations are demonstrated not only on canonical problems but also on realistic cases involving nanowires

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

combined field integral equation formulation for inhomogeneous two and three dimensional bodies the junction problem a combined field integral equation cfie formulation is presented for two and three dimensional bodies having discrete dielectric and conducting regions a three dimensional case is restricted to bodies of revolution bors the two dimensional case is analogous to the bor case when the fourier mode number is zero the method of moments mm is used to solve the cfie in terms of two integral operators it is shown that the cfie formulation yields accurate answers for scattering problems where the scatterer may be internally resonant the cfie results were validated using mie series results and measured data the junction problem associated with the cfie based formulation is explicated and several geometries with multiple junctions are used to validate the general cfie formulation a number of configurations are tested where the penetrable region consisted of a free space coating extensive numerical studies have shown that such limiting cases are sensitive indicators of the stability of mm solutions and allow direct comparison of different configurations

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

a characterization of compact convex polyhedra in hyperbolic 3 space in this paper we study the extrinsic geometry of convex polyhedral surfaces in three dimensional hyperbolic spaceh 3 we obtain a number of new uniqueness results and also obtain a characterization of the shapes of convex polyhedra inh 3 in terms of a generalized gauss map this characterization greatly generalizes andre ev s theorem

stabilization of integral equation formulations for the accurate solution of scattering problems involving low contrast dielectric objects the solution of scattering problems involving low contrast dielectric objects with three dimensional arbitrary shapes is considered using the traditional forms of the surface integral equations scattered fields cannot be calculated accurately if the contrast of the object is low therefore we consider the stabilization of the formulations by extracting the nonradiating parts of the equivalent currents we also investigate various types of stable formulations and show that accuracy can be improved systematically by eliminating the identity terms from the integral equation kernels traditional and stable formulations are compared not only for small scatterers but also for relatively large problems solved by employing the multilevel fast multipole algorithm stable and accurate solutions of dielectric contrasts as low as 10 4 are demonstrated on problems involving more than 250000 unknowns

a higher order multilevel fast multipole algorithm for scattering from mixed conducting dielectric bodies a higher order multilevel fast multipole algorithm mlfma is presented for computing electromagnetic scattering from three dimensional bodies comprising both conducting and dielectric objects the problem is formulated using the poggio miller chang harrington wu tsai 1973 1977 approach for multiple homogeneous dielectric objects and the combined field approach for conducting objects the resultant integral equations are discretized by the method of moments mom in which the conducting and dielectric surfaces interfaces are represented by curvilinear triangular patches and the unknown equivalent electric and magnetic currents are expanded using higher order vector basis functions such a discretization yields a highly accurate representation of the unknown currents without compromising the accuracy of geometrical modeling an implicit matrix filling scheme is employed to facilitate the treatment of complex scatterers having multiple junctions the resultant numerical system is then solved by mlfma which is tailored to accommodate the material properties of dielectric scatterers and the solution is accelerated using an incomplete lu decomposition preconditioner numerical examples are presented to demonstrate the accuracy and versatility of this approach in dealing with a wide array of scattering problems

piecewise linear petrov galerkin error estimates for the box method the perpendicular bisector box method pbm discretization of the laplacian on a simplicial element is equivalent to a piecewise linear petrov galerkin method with piecewise constant test functions this pbm element matrix is expressed in terms of galerkin s piecewise linear plg element matrix and the average values assumed by test functions over element faces only these results are shown to be applicable to the accuracy analysis of pbm discretizations of extremum stable elliptic gradient equations subject to regularity conditions on variable coefficients in these equations these gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on n dimensional delaunay triangulations this conclusion need not hold for the plg equations the dependence of the discretized laplacian on only the element face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions the latter are used to establish piecewise linear accuracy for the pbm by an adaptation of an error estimate for petrov galerkin methods by strang

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

on the evaluation of the 4 d reaction integral for the scalar potential in galerkin s method of moments despite the great progress made in the evaluation of the 4 d reaction integrals arising in the method of moments applied to the surface integral equation formulations there are still some aspects that deserve to be explored further the focus in this paper lies in the evaluation of the 4 d reaction integral for the scalar potential we demonstrate that unless this integral is computed to machine precision in the whole impedance matrix it is imperative to maintain the trianglewise balance in the evaluation of the scalar potential part a small imbalance even at very small error results in a loss of degrees of freedom dofs for the charge distribution violating the null space property of the divergence operator and yielding potentially spurious solutions the effect although present may go unnoticed at the conventional frequency regimes where the usual mesh sizes can be applied however it becomes more pronounced as we approach the quasi static limit eventually rendering spurious solutions at the frequencies higher than the classical low frequency breakdown would

a procedure for modeling material junctions in 3 d surface integral equation approaches surface integral equations become cumbersome to solve when arbitrary combinations of materials and complicated arrangements of junctions between those materials are considered this paper describes a straightforward approach for generalizing integral equation techniques to handle any combinations of materials with junctions particular attention is focused on a unique easily implemented junction resolution algorithm that maps each basis function coefficient to one or more unknowns three examples are presented for validation purposes

characterizations of circle patterns and finite convex polyhedra in hyperbolic 3 space the aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space mathbb h 3 we characterize the quasiconformal deformation space of each finite convex polyhedron as a corollary we obtain some results on finite circle patterns in the riemann sphere with dihedral angle 0 le theta pi that is for any circle pattern on hat mathbb c its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

on the convergence of circle packings to the riemann map rodin and sullivan 1987 proved thurston s conjecture that a scheme based on the circle packing theorem converges to the riemann mapping thereby providing a refreshing geometric view of riemann s mapping theorem we now present a new proof of the rodin sullivan theorem this proof is based on the argument principle and has the following virtues 1 it applies to more general packings the rodin sullivan paper deals with packings based on the hexagonal combinatorics later quantitative estimates were found which also worked for bounded valence packings here the bounded valence assumption is unnecessary and irrelevant 2 our method is rather elementary and accessible to non experts in particular quasiconformal maps are not needed consequently this gives an independent proof of riemann s conformal mapping theorem the rodin sullivan proof uses results that rely on riemann s mapping theorem 3 our approach gives the convergence of the first and second derivatives without significant additional difficulties while previous work has established the convergence of the first two derivatives for bounded valence packings now the bounded valence assumption is unnecessary

a conservative and monotone mixed hybridized finite element approximation of transport problems in heterogeneous domains in this article we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties the model in each subdomain consists of a partial differential equation with diffusive convective and reactive terms the coupling between each subdomain being realized through an interface transmission condition of robin type the numerical approximation of the problem in the two dimensional case is carried out through a dual mixed hybridized finite element method with numerical quadrature of the mass flux matrix the resulting method is a conservative finite volume scheme over triangular grids for which a discrete maximum principle is proved under the assumption that the mesh is of delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface the stability accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology electrophysiology and neuroelectronics

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

rigidity of infinite circle packings the nerve of a packing is a graph that encodes its combinatorics the vertices of the nerve correspond to the packed sets and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect