the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

hp dgfem for second order elliptic problems in polyhedra ii exponential convergence the goal of this paper is to establish exponential convergence of hp version interior penalty ip discontinuous galerkin dg finite element methods for the numerical approximation of linear second order elliptic boundary value problems with homogeneous dirichlet boundary conditions and piecewise analytic data in three dimensional polyhedral domains more precisely we shall analyze the convergence of the hp ip dg methods considered in d schotzau c schwab t p wihler siam j numer anal 51 2013 pp 1610 1633 based on axiparallel sigma geometric anisotropic meshes and bm s linear anisotropic polynomial degree distributions

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

on the approximability of numerical taxonomy fitting distances by tree metrics we consider the problem of fitting an n n distance matrix d by a tree metric t let varepsilon be the distance to the closest tree metric under the l infty norm that is varepsilon min t parallel t d parallel infty first we present an o n2 algorithm for finding a tree metric t such that parallel t d parallel infty leq 3 varepsilon second we show that it is cal np hard to find a tree metric t such that parallel t d parallel infty frac 9 8 varepsilon this paper presents the first algorithm for this problem with a performance guarantee

weighted analytic regularity in polyhedra we explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems the abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors paper in m costabel m dauge s nicaise analytic regularity for linear elliptic systems in polygons and polyhedra math models methods appl sci 22 8 2012 we illustrate this strategy by considering problems set in smooth domains in corner domains and in polyhedra

efficient algorithms for inverting evolution evolution can be mathematically modelled by a stochastic process that operates on the dna of species such models are based on the established theory that the dna sequences or genomes of all extant species have been derived from the genome of the common ancestor of all species by a process of random mutation and natural selection a stochastic model of evolution can be used to construct phylogenies or evolutionary trees for a set of species maximum likelihood estimation mle methods seek the evolutionary tree which is most likely to have produced the dna under consideration while these methods are intellectually satisfying they have not been widely accepted because of their computational intractability in this paper we address the intractability of mle methods as follows we introduce a metric on stochastic process models of evolution we show that this metric is meaningful by proving that in order for any algorithm to distinguish between two stochastic models that are close according to this metric it needs to be given many observations we complement this result with a simple and efficient algorithm for inverting the stochastic process of evolution that is for building a tree from observations on two state characters we will use the same techniques in a subsequent paper to solve the problem for multistate characters and hence for building a tree from dna sequence data the tree we build is provably close in our metric to the tree generating the data and gets closer as more observations become available though there have been many heuristics suggested for the problem of finding good approximations to the most likely tree our algorithm is the first one with a guaranteed convergence rate and further this rate is within a polynomial of the lower bound rate we establish ours is also the first polynomial time algorithm that is proven to converge at all to the correct tree

exponential convergence of mixed hp dgfem for stokes flow in polygons we analyze mixed hp discontinuous galerkin finite element methods dgfem for stokes flow in polygonal domains in conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders we prove that the hp dgfem leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corners

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

exponential convergence of the hp dgfem for diffusion problems abstract we prove exponential convergence of the hp version of discontinuous galerkin fem on geometrically refined meshes in polygons several variants of interior penalization are covered numerical experiments indicate the sharpness of the theoretical results as well as the weak dependence of the dgfem approximation on the particular choice of interior penalization and the penalty parameter

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

approximation properties of the h p version of the finite element method abstract the paper surveys the approximation and convergence properties of the h p version of the finite element method the basic theoretical results together with the main ideas of the proofs numerical examples are given

a few logs suffice to build almost all trees part ii abstract inferring evolutionary trees is an interesting and important problem in biology but one that is computationally difficult as most associated optimization problems are np hard although many methods are provably statistically consistent i e the probability of recovering the correct tree converges to 1 as the sequence length increases the actual rate of convergence for different methods has not been well understood in a recent paper we introduced a new method for reconstructing evolutionary trees called the dyadic closure method dcm and we showed that dcm has a very fast convergence rate dcm runs in o n 5 log n time where n is the number of sequences and so although polynomial the computational requirements are potentially too large to be of use in practice in this paper we present another tree reconstruction method the witness antiwitness method wam wam is faster than dcm especially on random trees and converges to the true tree topology at the same rate as dcm we also compare wam to other methods used to reconstruct trees including neighbor joining possibly the most popular method among molecular biologists and new methods introduced in the computer science literature

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

asymptotic spreading fastened by inter specific coupled nonlinearities a cooperative system abstract this paper is concerned with the asymptotic spreading of a lotka volterra cooperative system by using the theory of asymptotic spreading of nonautonomous equations the asymptotic speeds of spreading of unknown functions formulated by a coupled system are estimated our results imply that the asymptotic spreading of one species can be significantly fastened by introducing a mutual species which indicates the role of cooperation described by the coupled nonlinearities

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3 part i countably normed spaces on polyhedral domains this is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains and it concentrates on exploring the structure of these spaces such as the embeddings of weighted sobolev spaces the relation between weighted sobolev spaces and weighted continuous function spaces and the relations between the weighted sobolev spaces and countably weighted sobolev spaces in cartesian coordinates and in the spherical and cylindrical coordinates these well defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in 3 which are essential for the design of effective computation and the analysis of the h p version of the finite element method for solving elliptic problems in three dimensional nonsmooth domains arising from mechanics and engineering

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

provably fast and accurate recovery of evolutionary trees through harmonic greedy triplets we give a greedy learning algorithm for reconstructing an evolutionary tree based on a certain harmonic average on triplets of terminal taxa after the pairwise distances between terminal taxa are estimated from sequence data the algorithm runs in smallbigo numtaxa 2 time using smallbigo numtaxa work space where numtaxa is the number of terminal taxa these time and space complexities are optimal in the sense that the size of an input distance matrix is numtaxa 2 and the size of an output tree is numtaxa moreover in the jukes cantor model of evolution the algorithm recovers the correct tree topology with high probability using sample sequences of length polynomial in 1 numtaxa 2 the logarithm of the error probability and 3 the inverses of two small parameters

least squares spectral element preconditioners for fourth order elliptic problems abstract in this paper we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions d 2 3 using spectral element methods these preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert for second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least squares methods we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them numerical results for the condition number reflects the effectiveness of the preconditioners

h p spectral element methods for three dimensional elliptic problems on non smooth domains part iii error estimates preconditioners computational techniques and numerical results abstract the present paper is the third of a series of papers devoted to the study of h p spectral element methods for three dimensional elliptic problems on non smooth domains using parallel computers in this paper we provide error estimates preconditioners and numerical results the spectral element functions are fully non conforming we propose preconditioners on non smooth domains which can be diagonalized using separation of variables technique optimal error estimates in terms of number of layers in the geometrical mesh and in terms of number of degrees of freedom are obtained the method is easy to implement on a parallel computer and we briefly outline computational techniques we give results of numerical simulations to confirm the theoretical estimates theoretical results have been also validated by computational experiments which are published independently in dutt et al 2014

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

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

convergence and optimality of mathbf hp hp afem we design and analyze an adaptive hp finite element method mathbf hp hp afem in dimensions n 1 2 n 1 2 the algorithm consists of iterating two routines mathbf hp hp nearbest finds a near best hp approximation of the current discrete solution and data to a desired accuracy and reduce improves the discrete solution to a finer but comparable accuracy the former hinges on a recent algorithm by binev for adaptive hp approximation and acts as a coarsening step we prove convergence and instance optimality

mixed hp dgfem for incompressible flows we consider several mixed discontinuous galerkin approximations of the stokes problem and propose an abstract framework for their analysis using this framework we derive a priori error estimates for hp approximations on tensor product meshes we also prove a new stability estimate for the discrete divergence bilinear form