Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.     

Quantum codes from cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In particular, we decompose a cyclic code over \(F_q+uF_q+vF_q+uvF_q\) into four cyclic codes over \(F_q\) to determine the parameters of the corresponding quantum code.     

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

A Spectral Collocation Method for Nonlinear Fractional Boundary Value Problems with a Caputo Derivative

In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order \(\alpha \in (1,2)\) on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the \(L^2\)- and \(L^\infty \)-norms. Numerical experiments are included to illustrate the theoretical results.     

Local Discontinuous Galerkin Method for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in \(L^{\infty }(0, T; L^{2})\) for concentration c, \(L^{2}(0, T; L^{2})\) for \(\nabla c\) and \(L^{\infty }(0, T; L^{2})\) for velocity \(\mathbf{u}\) are derived. The main techniques in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments are shown to demonstrate the theoretical results.     

A Hybridized Discontinuous Galerkin Method with Reduced Stabilization

In this paper, we propose a hybridized discontinuous Galerkin (HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree \(k\) and \(k-1\) for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and \(L^2\) norms under the chunkiness condition. In the case of \(k=1\), it can be shown that the proposed method is closely related to the Crouzeix–Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.     

A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are \(L^2\) stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal \(L^2\) error estimate of \(O(h^{k+1})\) for polynomials of degree k for semi-discrete DG schemes, and the \(L^2\) error of \(O(h^{k+1} +(\Delta t)^2)\) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented.     

Efficient parallelizable hashing using small non-compressing primitives

A well-established method of constructing hash functions is to base them on non-compressing primitives, such as one-way functions or permutations. In this work, we present \(S^r\), an \(rn\)-to-\(n\)-bit compression function (for \(r\ge 1\)) making \(2r-1\) calls to \(n\)-to-\(n\)-bit primitives (random functions or permutations). \(S^r\) compresses its inputs at a rate (the amount of message blocks per primitive call) up to almost 1/2, and it outperforms all existing schemes with respect to rate and/or the size of underlying primitives. For instance, instantiated with the \(1600\)-bit permutation of NIST’s SHA-3 hash function standard, it offers about \(800\)-bit security at a rate of almost 1/2, while SHA-3-512 itself achieves only \(512\)-bit security at a rate of about \(1/3\). We prove that \(S^r\) achieves asymptotically optimal collision security against semi-adaptive adversaries up to almost \(2^{n/2}\) queries and that it can be made preimage secure up to \(2^n\) queries using a simple tweak.     

Fast L1–L2 Minimization via a Proximal Operator

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)–\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)–\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)–\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)–\(L_2\) based on a difference-of-convex approach in the numerical experiments.     

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.     

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order \(\mathcal{O}(\tau ^{r}+h^{4})\), where \(r\ge 2\) is a positive integer and \(\tau \) and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of \(2\le r\le 6\), and the optimal error estimates in the weighted \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.     

Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables \(S_i\) (\(=\pm \)1) and Z(2) synaptic connection (gauge) variables \(J_{ij}\) (\(=\pm \)1). Its energy consists of the Hopfield term \(-c_1S_iJ_{ij}S_j\), double Hopfield term \(-c_2 S_iJ_{ij}J_{jk} S_k\), and the reverberation (triple Hopfield) term \(-c_3 J_{ij}J_{jk}J_{ki}\) of synaptic self interactions. For the case \(c_2=0\), its phase diagram in the \(c_3-c_1\) plane has been studied both for the symmetric couplings \(J_{ij}=J_{ji}\) and asymmetric couplings (\(J_{ij}\) and \(J_{ji}\) are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing \(c_2\), and examine its effect. We find that the \(c_2\) term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of \(S_i\) and recalling them and measure the performance directly by overlaps of \(S_i\). We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.     

A locking-free stabilized mixed finite element method for linear elasticity: the high order case

In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\)-order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\)-norm and for the displacement in \(L^2\)-norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.     

No nexiste nce of n-qubit u nexte ndible product bases of size $$2^ n-5$$2 n-5

It is known that the n-qubit system has no unextendible product bases (UPBs) of cardinality \(2^n-1\), \(2^n-2\) and \(2^n-3\). On the other hand, the n-qubit UPBs of cardinality \(2^n-4\) exist for all \(n\ge 3\). We prove that they do not exist for cardinality \(2^n-5\).     

Multilevel quantum Otto heat engines with identical particles

A quantum Otto heat engine is studied with multilevel identical particles trapped in one-dimensional box potential as working substance. The symmetrical wave function for Bosons and the anti-symmetrical wave function for Fermions are considered. In two-particle case, we focus on the ratios of \(W^i\) (\(i=B,F\)) to \(W_s\), where \(W^\mathrm{B}\) and \(W^\mathrm{F}\) are the work done by two Bosons and Fermions, respectively, and \(W_s\) is the work output of a single particle under the same conditions. Due to the symmetrical of the wave functions, the ratios are not equal to 2. Three different regimes, low-temperature regime, high-temperature regime, and intermediate-temperature regime, are analyzed, and the effects of energy level number and the differences between the two baths are calculated. In the multiparticle case, we calculate the ratios of \(W^i_M/M\) to \(W_s\), where \(W^i_M/M\) can be seen as the average work done by a single particle in multiparticle heat engine. For other working substances whose energy spectrum has the form of \(E_n\sim n^2\), the results are similar. For the case \(E_n\sim n\), two different conclusions are obtained.     

Recursive Computation of Logarithmic Derivatives,Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (\(P_n^m(x)\), \(Q_n^m(x)\), \(n\ge m\ge 0\), \(x>1\)), the oblate spheroidal harmonics (\(P_n^m(ix)\), \(Q_n^m(ix)\), \(n\ge m\ge 0\), \(x>0\)), and the modified Bessel functions (\(I_n(x)\), \(K_n(x)\), \(n\ge 0\), \(x>0\)) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

Superconvergence of Immersed Finite Volume Methods for One-Dimensional Interface Problems

In this paper, we introduce a class of high order immersed finite volume methods (IFVM) for one-dimensional interface problems. We show the optimal convergence of IFVM in \(H^1\)- and \(L^2\)-norms. We also prove some superconvergence results of IFVM. To be more precise, the IFVM solution is superconvergent of order \(p+2\) at the roots of generalized Lobatto polynomials, and the flux is superconvergent of order \(p+1\) at generalized Gauss points on each element including the interface element. Furthermore, for diffusion interface problems, the convergence rates for IFVM solution at the mesh points and the flux at generalized Gauss points can both be raised to 2p. These superconvergence results are consistent with those for the standard finite volume methods. Numerical examples are provided to confirm our theoretical analysis.