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From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order P 1 − P 0 element for the Stokes region and P 1 element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the H 1 -norm for the velocity and piezometric head and in the L 2 -norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment. [ABSTRACT FROM AUTHOR]

In this paper, some results of a generalized fractional difference operator are studied. Fractional Euler difference sequence spaces e 0 t (B (α ~)) and e c t (B (α ~)) are introduced. Furthermore, some functional properties, Köthe-Toeplitz duals and continuous duals of these sequence spaces are examined. Finally, the definitions of fractional derivatives and integrals are introduced by applying the generalized difference scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in H1(RN)∩˙HSc(RN) and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded ˙HSc norm definitely have concentration properties related to a fixed ˙HSc norm of certain standing waves. [ABSTRACT FROM AUTHOR]

In this paper, we determine the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operator B v (m) on the sequence space ℓ p (1 < p < ∞). These results generalize the spectrum of the difference operator △ , △ v , △ a , b , △ u v 2 , △ u v w 2 , △ v r and some other cases Z s , B (r , s) , B (r , s , t) over the space ℓ p . [ABSTRACT FROM AUTHOR]

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank-Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank-Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes. [ABSTRACT FROM AUTHOR]

In this article, we construct an efficient numerical scheme to solve the coupled nonlinear Klein–Gordon system in a multi-dimensional unbounded domain R d (d = 1 , 2 , 3). We first use the SAV method to equivalently deform the original system, then we treat the nonlinear part of the system explicitly in temporal discretization. Thanks to the SAV method, we ensure the unconditional energy conservation of the discrete system, as a cost, we only need to solve two completely linearized decoupled systems. To avoid the errors caused by domain truncation and the construction of artificial boundary conditions, we apply the Hermite–Galerkin spectral method with scaling factor for spatial discretization. In order to save computational costs, we utilize two time adaptive strategies to adjust the time step size, through which we can improve computational efficiency without sacrificing accuracy. Finally, we presented numerical experiments to demonstrate the accuracy, efficiency, and energy conservation properties of our algorithm, and applied it to simulate the interactions of two-dimensional and three-dimensional vector solitons. [ABSTRACT FROM AUTHOR]