The Nonconvergence of h-Refinement in Prolate Elements

Prolate elements are a “plug-compatible” modification of spectral elements in which Legendre polynomials are replaced by prolate spheroidal wave functions of order zero. Prolate functions contain a“bandwidth parameter” $c \ge 0 $ c ≥ 0 whose value is crucial to numerical performance; the prolate functions reduce to Legendre polynomials for $c\,=\,0$ c = 0 . We show that the optimal bandwidth parameter $c$ c not only depends on the number of prolate modes per element $N$ N , but also on the element widths $h$ h . We prove that prolate elements lack $h$ h -convergence for fixed $c$ c in the sense that the error does not go to zero as the element size $h$ h is made smaller and smaller. Furthermore, the theoretical predictions that Chebyshev and Legendre polynomials require $\pi $ π degrees of freedom per wavelength to resolve sinusoidal functions while prolate series need only 2 degrees of freedom per wavelength are asymptotic limits as $N \rightarrow \infty $ N → ∞ ; we investigate the rather different behavior when $N \sim O(4-10)$ N ～ O ( 4 ? 10 ) as appropriate for spectral elements and prolate elements. On the other hand, our investigations show that there are certain combinations of $N,\,h$ N , h and $c>0$ c > 0 where a prolate basis clearly outperforms the Legendre polynomial approximation.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the $L^2$ stability of the scheme and several optimal $L^2$ error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are $\mathcal O (h^{k+3/2})$ super close to particular projections of the exact solutions for $k$ th-degree polynomial spaces while computational results show higher $\mathcal O (h^{k+2})$ convergence rate. Our proofs are valid for arbitrary regular meshes and for $P^k$ polynomials with $k\ge 1$ , and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the $L^2$ -norm under mesh refinement.     

The Compact Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

A compact discontinuous Galerkin method (CDG) is devised for nearly incompressible linear elasticity, through replacing the global lifting operator for determining the numerical trace of stress tensor in a local discontinuous Galerkin method (cf. Chen et al., Math Probl Eng 20, 2010) by the local lifting operator and removing some jumping terms. It possesses the compact stencil, that means the degrees of freedom in one element are only connected to those in the immediate neighboring elements. Optimal error estimates in broken energy norm, $H^1$ -norm and $L^2$ -norm are derived for the method, which are uniform with respect to the Lamé constant $\lambda .$ Furthermore, we obtain a post-processed $H(\text{ div})$ -conforming displacement by projecting the displacement and corresponding trace of the CDG method into the Raviart–Thomas element space, and obtain optimal error estimates of this numerical solution in $H(\text{ div})$ -seminorm and $L^2$ -norm, which are uniform with respect to $\lambda .$ A series of numerical results are offered to illustrate the numerical performance of our method.     

A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

A $C^0$ -weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for $C^0$ functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete $H^2$ norm and the standard $H^1$ and $L^2$ norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order $(k+1-d)$ and the surface mass of order $(k+2-d)$ for the $P_{k+2}$ finite element functions in $d$ -dimensional space.     

Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach $h^{r+2}$ and even $h^{2r}$ , comparing with $h^{r+1}$ rate of the counterpart finite element method. Here $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.     

Locality and checkability in wait-free computing

This paper studies notions of locality that are inherent to the specification of distributed tasks by identifying fundamental relationships between the various scales of computation, from the individual process to the whole system. A locality property called projection-closed is identified. This property completely characterizes tasks that are wait-free checkable, where a task $T =(\mathcal{I },\mathcal{O },\varDelta )$ T = ( I , O , Δ ) is said to be checkable if there exists a distributed algorithm that, given $s\in \mathcal{I }$ s ∈ I and $t\in \mathcal{O }$ t ∈ O , determines whether $t\in \varDelta {(s)}$ t ∈ Δ ( s ) , i.e., whether $t$ t is a valid output for $s$ s according to the specification of $T$ T . Projection-closed tasks are proved to form a rich class of tasks. In particular, determining whether a projection-closed task is wait-free solvable is shown to be undecidable. A stronger notion of locality is identified by considering tasks whose outputs “look identical” to the inputs at every process: a task $T= (\mathcal{I },\mathcal{O },\varDelta )$ T = ( I , O , Δ ) is said to be locality-preserving if $\mathcal{O }$ O is a covering complex of $\mathcal{I }$ I . We show that this topological property yields obstacles for wait-free solvability different in nature from the classical impossibility results. On the other hand, locality-preserving tasks are projection-closed, and thus they are wait-free checkable. A classification of locality-preserving tasks in term of their relative computational power is provided. This is achieved by defining a correspondence between subgroups of the edgepath group of an input complex and locality-preserving tasks. This correspondence enables to demonstrate the existence of hierarchies of locality-preserving tasks, each one containing, at the top, the universal task (induced by the universal covering complex), and, at the bottom, the trivial identity task.     

Variational Image Segmentation Models Involving Non-smooth Data-Fidelity Terms

This article introduces a class of piecewise-constant image segmentation models that involves $L^1$ norms as data fidelity measures. The $L^1$ norms enable to segment images with low contrast or outliers such as impulsive noise. The regions to be segmented are represented as smooth functions instead of the Heaviside expression of level set functions as in the level set method. In order to deal with both non-smooth data-fitting and regularization terms, we use the variable splitting scheme to obtain constrained optimization problems, and apply an augmented Lagrangian method to solve the problems. This results in fast and efficient iterative algorithms for piecewise-constant image segmentation. The segmentation framework is extended to vector-valued images as well as to a multi-phase model to deal with arbitrary number of regions. We show comparisons with Chan-Vese models that use $L^2$ fidelity measures, to enlight the benefit of the $L^1$ ones.     

Solving Convection-Diffusion Problems on Curved Domains by Extensions from Subdomains

We present a technique for numerically solving convection-diffusion problems in domains $\varOmega $ with curved boundary. The technique consists in approximating the domain $\varOmega $ by polyhedral subdomains $\mathsf{{D}}_h$ where a finite element method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain $\varOmega $ . This approach allows for the use of only polyhedral elements; there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. To achieve this, the boundary condition on the border of $\varOmega $ is transferred to the border of $\mathsf{D }_h$ by using simple line integrals. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that, whenever the distance of $\mathsf{{D}}_h$ to $\partial \varOmega $ is of order of the meshsize $h$ , the convergence properties of the resulting method are the same as those for the case in which $\varOmega =\mathsf{{D}}_h$ . We also show numerical evidence indicating that the ratio of the $L^2(\varOmega )$ norm of the error in the scalar variable computed with $d>0$ to that of that computed with $d=0$ remains constant (and fairly close to one), whenever the distance $d$ is proportional to $\min \{h,Pe^{-1}\}/(k+1)^2$ , where $Pe$ is the so-called Péclet number.     

Optimal quadrature formulas in the sense of Sard in W_2^{(m,m-1)} space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}(0,1)$ space. Using the Sobolev’s method we obtain new optimal quadrature formulas of such type for $N+1\ge m$ , where $N+1$ is the number of the nodes. Moreover, explicit formulas of the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$ and prove an asymptotic optimality of such a formula in the Sobolev space $L_2^{(1)}(0,1)$ . It turns out that the error of the optimal quadrature formula in $W_2^{(1,0)}(0,1)$ is less than the error of the optimal quadrature formula given in the $L_2^{(1)}(0,1)$ space. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}(0,1)$ space is exact for $\exp (-x)$ and $P_{m-2}(x)$ , where $P_{m-2}(x)$ is a polynomial of degree $m-2$ . Furthermore, some numerical results, which confirm the obtained theoretical results of this work, are given.     

Anisotropic Nonconforming { EQ}_1^{rot} Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

The main aim of this paper is to study the nonconforming $EQ_1^{rot}$ quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and $L^2$ -norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.     

Maximizing the weighted number of just-in-time jobs on a single machine with position-dependent processing times

We study the problem of maximizing the weighted number of just-in-time jobs on a single machine with position-dependent processing times. Unlike the vast majority of the literature, we do not restrict ourselves to a specific model of processing time function. Rather, we assume that the processing time function can be of any functional structure that is according to one of the following two cases. The first is the case where the job processing times are under a learning effect, i.e., each job processing time is a non-increasing function of its position in the sequence. In the second case, an aging effect is assumed, i.e., each job processing time is a non-decreasing function of its position in the sequence. We prove that the problem is strongly $\mathcal{N }\mathcal{P }$ N P -hard under a learning effect, even if all the weights are identical. When there is an aging effect, we introduce a dynamic programming (DP) procedure that solves the problem with arbitrary weights in $O(n^{3})$ O ( n 3 ) time (where $n$ n is the number of jobs). For identical weights, a faster optimization algorithm that runs in $O(n\log n)$ O ( n log n ) time is presented. We also extend the analysis to the case of scheduling on a set of $m$ m parallel unrelated machines and provide a DP procedure that solves the problem in polynomial time, given that $m$ m is fixed and that the jobs are under an aging effect.     

Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems

We consider charged transport within a porous medium, which at the pore scale can be described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system. We state three different homogenization results using the method of two-scale convergence. In addition to the averaged macroscopic equations, auxiliary cell problems are solved in order to provide closed-form expressions for effective coefficients. Our aim is to study numerically the convergence of the models for vanishing microstructure, i. e., the behavior for $\varepsilon \rightarrow 0$ ε → 0 , where $\varepsilon $ ε is the characteristic ratio between pore diameter and size of the porous medium. To this end, we propose a numerical scheme capable of solving the fully coupled microscopic SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time using mixed finite elements in two space dimensions. The averaged models are evaluated using simulation results and their approximation errors in terms of $\varepsilon $ ε are estimated numerically.     

Derandomized Parallel Repetition Theorems for Free Games

Raz’s parallel repetition theorem (SIAM J Comput 27(3):763–803, 1998) together with improvements of Holenstein (STOC, pp 411–419, 2007) shows that for any two-prover one-round game with value at most ${1- \epsilon}$ 1 - ? (for ${\epsilon \leq 1/2}$ ? ≤ 1 / 2 ), the value of the game repeated n times in parallel on independent inputs is at most ${(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}$ ( 1 - ? ) Ω ( ? 2 n ? ) , where ? is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352–365, 2009). Consequently, ${n=O(\frac{t \ell}{\epsilon})}$ n = O ( t ? ? ) repetitions suffice to reduce the value of a free game from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t , and denoting the input length of the game by m, it follows that ${nm=O(\frac{t \ell m}{\epsilon})}$ n m = O ( t ? m ? ) random bits can be used to prepare n independent inputs for the parallel repetition game. In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m + ?)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t while only multiplying the randomness complexity by O(t) when m = O(?). Our technique uses strong extractors to “derandomize” a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363–372, 1997) for communication games in the special case that the game is free.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

Interpolation splines minimizing a semi-norm

Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in $K_2(P_2)$ , where $K_2(P_2)$ is the space of functions $\phi $ such that $\phi ^{\prime } $ is absolutely continuous, $\phi ^{\prime \prime } $ belongs to $L_2(0,1)$ and $\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $ . Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the trigonometric functions $\sin x$ and $\cos x$ . Finally, in a few numerical examples the qualities of the defined splines and $D^2$ -splines are compared. Furthermore, the relationship of the defined splines with an optimal quadrature formula is shown.     

Simple characterization of strict residuated lattices with an involutive negation

We consider property of strict residuated lattices (SRL-algebras) with a new involutive negation $\lnot, $ called here by SRL $_{\lnot }$ -algebras, and give a simple characterization of SRL $_{\lnot }$ -algebras. We also prove a prime filter theorem of SRL $_{\lnot }$ -algebras, from which we conclude that every linearly ordered SRL $_{\lnot }$ -algebra is simple. As a corollary to this fact, we have a well-known result that every SML $_{\lnot }$ -algebra (SBL $_{\lnot }$ -algebra) is a subdirect product of linearly ordered SML $_{\lnot }$ -algebras (SBL $_{\lnot }$ -algebras).     

Reduced ordered binary decision diagram with implied literals: a new knowledge compilation approach

Reduced ordered binary decision diagram (ROBDD) is one of the most influential knowledge compilation languages. We generalize it by associating some implied literals with each node to propose a new language called ROBDD with implied literals (ROBDD- $L$ ) and show that ROBDD- $L$ can meet most of the querying requirements involved in the knowledge compilation map. Then, we discuss a kind of subsets of ROBDD- $L$ called ROBDD- $L_i$ with precisely $i$ implied literals $(0\le i\le \infty )$ , where ROBDD- $L_0$ is isomorphic to ROBDD. ROBDD- $L_i$ has uniqueness over any given linear order of variables. We mainly focus on ROBDD- $L_\infty $ and demonstrate that: (a) it is a canonical representation on any given variable order; (b) it is the most succinct subset in ROBDD- $L$ and thus also meets most of the querying requirements; (c) given any logical operation ROBDD supports in polytime, ROBDD- $L_\infty $ can also support it in time polynomial in the sizes of the equivalent ROBDDs. Moreover, we propose an ROBDD- $L_i$ compilation algorithm for any $i$ and an ROBDD- $L_\infty $ compilation algorithm, and then we implement an ROBDD- $L$ package called BDDjLu. Our preliminary experimental results indicate that: (a) the compilation results of ROBDD- $L_\infty $ are significantly smaller than those of ROBDD; (b) the standard d-DNNF compiler c2d and our ROBDD- $L_\infty $ compiler do not dominate the other, yet ROBDD- $L_\infty $ has canonicity and supports more querying requirements and relatively efficient logical operations; and (c) the method that first compiles knowledge base into ROBDD- $L_\infty $ and then converts ROBDD- $L_\infty $ into ROBDD provides an efficient ROBDD compiler.     

Distinguishing and relating higher-order and first-order processes by expressiveness

This is a paper on distinguishing and relating two important kinds of calculi through expressiveness, settling some critical but long unanswered questions. The delimitation of higher-order and first-order process calculi is a basic and pivotal topic in the study of process theory. Particularly, expressiveness studies mutual encodability, which helps decide whether process-passing or name-passing is more fundamental, and the way they ought to be used in both theory and practice. In this paper, we contribute to such demarcation with three major results. Firstly $\pi $ (first-order pi-calculus) can faithfully express $\varPi $ (basic higher-order pi-calculus). The calculus $\varPi $ has the elementary operators (input, output, composition and restriction). This actually is a corollary of a more general result, that $\pi $ can encode $\varPi ^r$ ( $\varPi $ enriched with the relabelling operator). Secondly $\varPi $ cannot interpret $\pi $ reasonably. This is of more significance since it separates $\varPi $ and $\pi $ by drawing a well-defined boundary. Thirdly an encoding from $\pi $ to $\varPi ^r$ is revisited and discussed, which not only implies how to make $\varPi $ more useful but also stresses the importance of name-passing in $\pi $ .     

General DG-Methods for Highly Indefinite Helmholtz Problems

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $\mathbb R ^{d}$ , $d\in \{1,2,3\}$ . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $hp$ -version of the finite element method explicitly in terms of the mesh width $h$ , polynomial degree $p$ and wavenumber $k$ . It is shown that the optimal convergence order estimate is obtained under the conditions that $kh/\sqrt{p}$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$ . On regular meshes, the first condition is improved to the requirement that $kh/p$ be sufficiently small.     

Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$ in standard $H^1$ norm, where $\tau $ is the temporal grid size and $h_1,h_2$ are spatial grid sizes; the other is $\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$ in $H^1_{\gamma }$ norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.