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

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

hp dgfem for second order elliptic problems in polyhedra i stability on geometric meshes we introduce and analyze hp version discontinuous galerkin dg finite element methods for the numerical approximation of linear second order elliptic boundary value problems in three dimensional polyhedral domains to resolve possible corner edge and corner edge singularities we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods similarly the local polynomial degrees are increased linearly and possibly anisotropically away from singularities we design interior penalty hp dg methods and prove that they are well defined for problems with singular solutions and stable under the proposed hp refinements we establish abstract error bounds that will allow us to prove exponential rates of convergence in the second part of this work

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

neural network approach to computing matrix inversion a neural network for matrix inversion is proposed we show both analytically and by simulations that this network is guaranteed to be stable and to provide results arbitrarily close to the accurate inversion of a matrix within an elapsed time of only a few characteristic time constants of the network the parameters such as the interconnection strengths of this network can be obtained from the known matrix without any computations

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

tree approximation for hp adaptivity the hp adaptive approximation is formulated as an approximation problem on a full binary tree t where for each of the leaves delta an order p delta ge1 is assigned in such a way that the

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

adaptive mesh strategies for the spectral element method abstract an adaptive spectral element method has been developed for the efficient solution of time dependent partial differential equations adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the one dimensional viscous burgers equation and the two dimensional navier stokes equations for driven flow in a cavity sharp gradients singularities and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used the adaptive formulation presents significant increases in efficiency flexibility and general capabilities for high order spectral methods

on p robust saturation for hp afem for the poisson problem in two dimensions we consider the standard adaptive finite element loop solve estimate mark refine with estimate being implemented using the p robust equilibrated flux estimator and mark being drfler marking as a refinement strategy we employ p refinement we investigate the question by which amount the local polynomial degree on any marked patch has to be incremented in order to achieve a p independent error reduction we show that the analysis can be transferred from the patches to a reference triangle and therein we provide clear cut computational evidence that any increment proportional to the polynomial degree for any fixed proportionality constant yields the desired reduction the resulting adaptive method can be turned into an instance optimal hp adaptive method by the addition of a coarsening routine

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

an adaptive hp refinement strategy with computable guaranteed bound on the error reduction factor we propose a new practical adaptive refinement strategy for hp finite element approximations of elliptic problems following recent theoretical developments in polynomial degree robust a posteriori error analysis we solve two types of discrete local problems on vertex based patches the first type involves the solution on each patch of a mixed finite element problem with homogeneous neumann boundary conditions which leads to an mathbf h mathrm div omega conforming equilibrated flux this in turn yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via dorfler s bulk chasing criterion the second type of local problems involves the solution on patches associated with marked vertices only of two separate primal finite element problems with homogeneous dirichlet boundary conditions which serve to decide between h p or hp refinement altogether we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements error reduction factor in a series of numerical experiments featuring smooth and singular solutions we study the performance of the proposed hp adaptive strategy and observe exponential convergence rates we also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction and we find that this ratio is in general quite close to the optimal value of one

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

contraction and optimality properties of an adaptive legendre galerkin method the multi dimensional case we analyze the theoretical properties of an adaptive legendre galerkin method in the multidimensional case after the recent investigations for fourier galerkin methods in a periodic box and for legendre galerkin methods in the one dimensional setting the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral hp hp discretizations of elliptic boundary value problems the main contribution of the paper is a careful construction of a multidimensional riesz basis in h 1 h1 based on a quasi orthonormalization procedure this allows us to design an adaptive algorithm to prove its convergence by a contraction argument and to discuss its optimality properties in the sense of non linear approximation theory in certain sparsity classes of gevrey type

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

a further three critical points theorem abstract if x is a real banach space we denote by w x the class of all functionals x r possessing the following property if u n is a sequence in x converging weakly to u x and lim inf n u n u then u n has a subsequence converging strongly to u in this paper we prove the following result let x be a separable and reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional belonging to w x bounded on each bounded subset of x and whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that for each i the functional j is coercive and has a strict local not global minimum say x then for each compact interval a b i for which sup a b x j x there exists r 0 with the following property for every a b and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x has at least three solutions whose norms are less than r

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

real time computation of the eigenvectors of a class of positive definite matrices this paper proposes an analog circuit approach to computing in real time the eigenvectors corresponding to all the eigenvalues of a class of positive definite matrices we show analytically and by simulations that the proposed circuit can provide the desired eigenvectors with arbitrarily small error during an elapsed time of only a few characteristic time constants of the circuit

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

adaptive fourier galerkin methods we study the performance of adaptive fourier galerkin methods in a periodic box in r d with dimension d 1 these methods offer unlimited approximation power only restricted by solution and data regularity they are of intrinsic interest but are also a first step towards understanding adaptivity for the hp fem we examine two nonlinear approximation classes one classical corresponding to algebraic decay of fourier coefficients and another associated with exponential decay typical of spectral approximation we investigate the natural sparsity class for the operator range and find that the exponential class is not preserved thus in contrast with the algebraic class this entails a striking different behavior of the feasible residuals that lead to practical algorithms influencing the overall optimality the sparsity degradation for the exponential class is partially compensated with coarsening we present several feasible adaptive fourier algorithms prove their contraction properties and examine the cardinality of the activated sets the galerkin approximations at the end of each iteration are quasi optimal for both classes but inner loops or intermediate approximations are sub optimal for the exponential class

exponential convergence in h 1 of hp fem for gevrey regularity with isotropic singularities for functions u in h 1 omega in a bounded polytope omega subset mathbb r d d 1 2 3 with plane sides for d 2 3 which are gevrey regular in overline omega backslash mathscr s with point singularities concentrated at a set mathscr s subset overline omega consisting of a finite number of points in overline omega we prove exponential rates of convergence of hp version continuous galerkin finite element methods on affine families of regular simplicial meshes in omega the simplicial meshes are geometrically refined towards mathscr s but are otherwise unstructured

existence of entire solutions for a class of variable exponent elliptic equations abstract the paper deals with the existence of entire solutions for a quasilinear equation e in r n depending on a real parameter which involves a general variable exponent elliptic operator a in divergence form and two main nonlinearities the competing nonlinear terms combine each other under some conditions we prove the existence of a critical value 0 with the property that e admits nontrivial nonnegative entire solutions if and only if furthermore under the further assumption that the potential a of a is uniform convex we give the existence of a second independent nontrivial nonnegative entire solution of e when our results extend the previous work of autuori and pucci 2013 6 from the case of constant exponents p q and r to the case of variable exponents more interesting we weaken the condition max 2 p q min r p to the simple request that 1 q r furthermore we extend the previous work of alama and tarantello 1996 2 from dirichlet laplacian problems in bounded domains of r n to the case of a general variable exponent differential equation in the entire r n and also remove the assumption q 2 hence the results of this paper are new even in the canonical case p 2

kirchhoff type problems on a geodesic ball of the hyperbolic space abstract in this paper we study the existence of weak solutions for some kirchhoff type problems whose simple prototype is given by a b b h u 2 d h u f u in b r u 0 on b r where h denotes the laplace beltrami operator on the ball model of the hyperbolic space b n with n 3 a b and are real parameters b r b n is a geodesic ball centered in zero of radius r and f is a subcritical continuous function the kirchhoff term is allowed to vanish at the origin covering the degenerate case the main technical approach is based on variational and topological methods

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

on well posedness of semilinear parabolic and elliptic problems in the hyperbolic space we investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n dimensional hyperbolic space n 3 lp lq estimates for the semigroup generated by the laplace beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems moreover under proper assumptions on the nonlinear function we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem furthermore for the corresponding semilinear parabolic problem nonuniqueness of weak solutions is stated

fast computation in adaptive tree approximation adaptive methods of approximation arise in many settings including numerical methods for pdes and image processing they can usually be described by a tree which records the adaptive decisions this paper is concerned with the fast computation of near optimal trees based on n adaptive decisions the best tree based on n adaptive decisions could be found by examining all such possibilities however this is exponential in n and could be numerically prohibitive the main result of this paper is to show that it is possible to find near optimal trees using computations linear in n

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

a posteriori error estimation and adaptivity in hp virtual elements an explicit and computable error estimator for the hp version of the virtual element method vem together with lower and upper bounds with respect to the exact energy error is presented such error estimator is employed to provide following the approach of melenk and wohlmuth adv comput math 15 1 4 311 331 2001 hp adaptive mesh refinements for very general polygonal meshes in addition a novel vem hp clement quasi interpolant instrumental for the a posteriori error analysis is introduced the performances of the adaptive method are validated by a number of numerical experiments

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

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