An Analysis of the Dissipation and Dispersion Errors of the P N P M Schemes

We examine the dispersion and dissipation properties of the P _N P _M schemes for linear wave propagation problems. P _N P _M scheme are based on P _N discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree N up to the design polynomial degree M. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (N=0) and discontinuous Galerkin schemes (N=M) are included as special cases. We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of N and M exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.     

The Inf-Sup Condition for the Mapped Q k ?P k?1 disc Element in Arbitrary Space Dimensions

 One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is the Q _k −P _k−1 ^disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the P _k−1 ^disc space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension. Received January 31, 2001; revised May 2, 2002 Published online: July 26, 2002     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

Optimization problems and the polynomial hierarchy

It is demonstrated that such problems as the symmetric Travelling Salesman Problem, Chromatic Number Problem, Maximal Clique Problem and a Knapsack Packing Problem are in the Δ^P₂ level of PH and no lower if ∑^P₁ ≠ Π^P₁, or NP≠co-NP. This shows that these problems cannot be solved by polynomial reductions that use only positive information from an NP oracle, if NP≠co-NP. It is then shown how to extend these results to prove that interesting problems are properly in Δ^P,X_i+1 for all X,k where ∑^P,X_k≠Π^P,X_k in PH^X.     

On linear interpolation under interval data

Some results related to the problem of interpolation of n vertical segments (x_k, Y_k), k = 1,…,n, in the plane with generalized polynomial functions that are linear combinations of m basic functions are presented. It is proved that the set of interpolating functions (if not empty) is bounded in every subinterval (x_k, x_k+1) by two unique such functions η_k⁻ and η_k⁺. An algorithm with result verification for the determination of the boundary functions η_k⁻, η_k⁺ and for their effective tabulation is reported and some examples are discussed.     

Polynomoperatoren in C k [a,b]

Some estimations for the relative projection constantP(∏ _n+k ^k ,Csik[a,b]) are given. By constructing an associated polynomial operatorL _n :C ⁰[a,b]→∏ _n ⁰ to a given polynomial operatorH _n+k :C ^k [a,b]→∏ _n+k ^k we get a lower bound for the projection constant. An upper bound forP(∏ _n+k ^k ,C ^k [a,b]) is obtained by the determination of the norms of appropriate polynomial operatorsP _n+k :C ^k [a,b]→∏ _n+k ^k . Further we give a convergence property for the sequence (P _n+k ) _n∈?.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

Divergence-Free HDG Methods for the Vorticity-Velocity Formulation of the Stokes Problem

We study a hybridizable discontinuous Galerkin method for solving the vorticity-velocity formulation of the Stokes equations in three-space dimensions. We show how to hybridize the method to avoid the construction of the divergence-free approximate velocity spaces, recover an approximation for the pressure and implement the method efficiently. We prove that, when all the unknowns use polynomials of degree k??0, the L ² norm of the errors in the approximate vorticity and pressure converge with order k+1/2 and the error in the approximate velocity converges with order k+1. We achieve this by letting the normal stabilization function go to infinity in the error estimates previously obtained for a hybridizable discontinuous Galerkin method.     

Some results on graphs without long induced paths

Many NP-hard graph problems remain difficult on P_k-free graphs for certain values of k. Our goal is to distinguish subclasses of P_k-free graphs where several important graph problems can be solved in polynomial time. In particular, we show that the independent set problem is polynomial-time solvable in the class of (P_k,K_1,n)-free graphs for any positive integers k and n, thereby generalizing several known results.     

The Space Complexity of Approximating the Frequency Moments

The frequency moments of a sequence containingm_ielements of typei, 1⩽i⩽n, are the numbersF_k=∑ⁿ_i=1 m^k_i. We consider the space complexity of randomized algorithms that approximate the numbersF_k, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF₀,F₁, andF₂can be approximated in logarithmic space, whereas the approximation ofF_kfork⩾6 requiresn^Ω(1)space. Applications to data bases are mentioned as well.     

Verifiable quantum (k, n)-threshold secret sharing

In a conventional quantum (k, n) threshold scheme, a trusted party shares a quantum secret with n agents such that any k or more agents can cooperate to recover the original secret, while fewer than k agents obtain no information about the secret. Is the reconstructed quantum secret same with the original one? Or is the dishonest agent willing to provide a true share during the secret reconstruction? In this paper we reexamine the security of quantum (k, n) threshold schemes and show how to construct a verifiable quantum (k, n) threshold scheme by combining a qubit authentication process. The novelty of ours is that it can provide a mechanism for checking whether the reconstructed quantum secret is same with the original one. This mechanism can also attain the goal of checking whether the dishonest agent provides a false quantum share during the secret reconstruction such that the secret quantum state cannot be recovered correctly.     

Improvement of the accuracy in boundary element method based on high-order discretization

The boundary element method (BEM) is a popular method to solve various problems in engineering and physics and has been used widely in the last two decades. In high-order discretization the boundary elements are interpolated with some polynomial functions. These polynomials are employed to provide higher degrees of continuity for the geometry of boundary elements, and also they are used as interpolation functions for the variables located on the boundary elements. The main aim of this paper is to improve the accuracy of the high-order discretization in the two-dimensional BEM. In the high-order discretization, both the geometry and the variables of the boundary elements are interpolated with the polynomial function P_m, where m denotes the degree of the polynomial. In the current paper we will prove that if the geometry of the boundary elements is interpolated with the polynomial function P_m+1 instead of P_m, the accuracy of the results increases significantly. The analytical results presented in this work show that employing the new approach, the order of convergence increases from O(L₀)^m to O(L₀)^m+1 without using more CPU time where L₀ is the length of the longest boundary element. The theoretical results are also confirmed by some numerical experiments.     

Adjoint Recovery of Superconvergent Linear Functionals from Galerkin Approximations. The One-dimensional Case

In this paper, we extend the adjoint error correction of Pierce and Giles (SIAM Rev. 42, 247–264 (2000)) for obtaining superconvergent approximations of functionals to Galerkin methods. We illustrate the technique in the framework of discontinuous Galerkin methods for ordinary differential and convection–diffusion equations in one space dimension. It is well known that approximations to linear functionals obtained by discontinuous Galerkin methods with polynomials of degree k can be proven to converge with order 2k + 1 and 2k for ordinary differential and convection–diffusion equations, respectively. In contrast, the order of convergence of the adjoint error correction method can be proven to be 4k + 1 and 4k, respectively. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. Numerical results which confirm the theoretical predictions are presented.     

Optimal Error Estimates of the Local Discontinuous Galerkin Method for Surface Diffusion of Graphs on Cartesian Meshes

In (Xu and Shu in J. Sci. Comput. 40:375–390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L ² norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.     

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L₂-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.     

On γ-reducibility versus polynomial time many-one reducibility

We prove that each element of a class of functions (denoted by NPC_t^P), whose graphs can be accepted in nondeterministic polynomial time, can be evaluated in deterministic polynomial time if and only if γ-reducibility is equivalent to polynomial time many-one reducibility. We also modify the proof technique used to obtain part of this result to obtain the stronger result that if every γ-reduction can be replaced by a polynomial time Turing reduction, then every function in NPC_t^P can be evaluated in deterministic polynomial time.     

Complete problems for deterministic polynomial time

Log space reducibility allows a meaningful study of complexity and completeness for the class $\text{P}$ of problems solvable in polynomial time (as a function of problem size). If any one complete problem for $\text{P}$ is recognizable in log^k(n) space (for a fixed k), or requires at least n^c space (where c depends upon the program), then all complete problems in $\text{P}$ have the same property. A variety of familiar problems are shown complete for $\text{P}$, including context-free emptiness, infiniteness and membership, establishing inconsistency of propositional formulas by unit resolution, deciding whether a player in a two-person game has a winning strategy, and determining whether an element is generated from a set by a binary operation.     

Learning Rates for Regularized Classifiers Using Trigonometric Polynomial Kernels

Regularized classifiers are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the trigonometric polynomial kernels are ones of the most important kernels and play key roles in signal processing. The main target of this paper is to provide convergence rates of classification algorithms generated by regularization schemes with trigonometric polynomial kernels. As a special case, an error analysis for the support vector machines (SVMs) soft margin classifier is presented. The norms of Fejér operator in reproducing kernel Hilbert space and properties of approximation of the operator in L ¹ space with periodic function play key roles in the analysis of regularization error. Some new bounds on the learning rate of regularization algorithms based on the measure of covering number for normalized loss functions are established. Together with the analysis of sample error, the explicit learning rates for SVM are also derived.