Search

It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown.

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed.

We consider a Beris-Edwards system modeling incompressible liquid crystal flows of nematic type. This system couples a Navier–Stokes system for the fluid velocity with a time-dependent system for the Q-tensor variable, whose spectral decomposition is related to the directors of liquid crystal molecules. The long-time behavior for global weak solutions is studied, proving that each whole trajectory converges to a single equilibrium whenever a regularity hypothesis is satisfied by the energy of the weak solution.

Several kinds of approximation of the gravitational potential of a homogeneous body by truncated spherical harmonics series are in use in physical geodesy. However, only one of them is capable of a representation converging to the true potential in the whole layer between the Brillouin sphere and the Bjerhammar sphere of the body. We aim at providing various majorizations, namely upper bounds, of the error with the double purpose of proving explicitly the convergence in the sense of different norms and of giving computable bounds, that might be used in numerical studies. The first aim is reached for all the norms. For the second, however, it turns out that among the bounds, when applied to the example of the terrain correction of the Earth, only those referring to the mean absolute error and the mean squared error at the level of Brillouin sphere of minimum radius give significant and useful results. In order to make the computation an easy exercise, a simple approximate formula has been developed requiring only the use of the distribution function of the heights of the surface of the body with respect to the Bjerhammar sphere.

A mathematical model which describes a frictional contact problem between a piezoelectric body and an electrically conductive foundation is considered. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky’s law. The contact is modeled with Signorini’s conditions, a version of Coulomb’s law in which the coefficient of friction depends on the slip and a regularized electrical conductivity condition. A variational formulation for the problem is derived; then, the existence of a unique weak solution to the model is proved. Afterward, to solve the problem numerically, a successive iteration technique is proposed, and its convergence is established. Then, a variant of the augmented Lagrangian, the so-called alternating direction method of multipliers, is used to decompose the original problem into two sub-problems, solve them sequentially and update the dual variables at each iteration. Finally, to study the influence of the foundation’s conductivity on the iterative process, numerical experiments of two-dimensional test problems are carried out.

This paper proposes weak-form differential quadrature finite elements for strain gradient functionally graded (FG) Euler–Bernoulli and Timoshenko micro-beams. The elements developed both have six degrees of freedom per node and do not require shape functions. The effective material properties are assumed to change continuously along the thickness direction. To guarantee the inter-element continuity conditions, we construct sixth- and fourth-order differential quadrature-based geometric mapping schemes. The two mapping schemes are combined with the minimum potential energy principle to derive their respective element formulations. Several illustrative examples are presented to demonstrate the convergence and adaptability of our elements. Finally, we utilize the latter element to explore the size-dependent vibration characteristics of multiple-stepped FG micro-beams. Numerical results reveal that our elements have distinct convergence and adaptability advantages over the related standard finite elements. The step location, thickness ratio, power-law index, and material length scale parameter have notable impacts on the structural vibration frequencies and mode shapes.

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree no greater than 1. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive t, uniform (with respect to t) convergence to zero as |x| → ∞, and vanishing (for any x) starting with a certain sufficiently large t. For the singular case, we proved that the problem considered has no global positive solutions.

In this paper, the coupled phase field and local/nonlocal integral elasticity theories are used for stress-induced martensitic phase transformations (MPTs) at the nanoscale to investigate the limitations and contradictions of the nonlocal integral elasticity, which are due to the fact that the support of the nonlocal kernel exceeds the integration domain, i.e., the boundary effect. Different functions for the nonlocal kernel are compared. In order to compensate the boundary effect, a new nonlocal kernel, i.e., the compensated two-phase kernel, is introduced, in which a local part is added to the nonlocal part of the two-phase kernel to account for the boundary effect. In contrast to the previously introduced modified kernel, the compensated two-phase kernel does not lead to a purely nonlocal behavior in the core region, and hence no singular behavior, and consequently, no computational convergence issue is observed. The nonlinear finite element approach and the COMSOL code are used to solve the coupled system of Ginzburg–Landau and local/nonlocal integral elasticity equations. The numerical implementation of the phase field-local elasticity equations and the 2D nonlocal integral elasticity are verified. Boundary effect is investigated for MPT with both homogeneous and nonhomogeneous stress distributions. For the former, in contrast to the local elasticity, a nonhomogeneous phase transformation (PT) occurs in the nonlocal case with the two-phase kernel. Using the compensated two-phase kernel results in a homogeneous PT similar to the local elasticity. For the latter, the sample transforms to martensite except the adjacent region to the boundary for the local elasticity, while for the two-phase kernel, the entire sample transforms to martensite. The solution of the compensated two-phase kernel, however, is very similar to that of the local elasticity. The applicability of boundary symmetry in phase field problems is also investigated, which shows that it leads to incorrect results within the nonlocal integral elasticity. This is because when the symmetric portions of a sample are removed, the corresponding nonlocal effects on the remaining portion are neglected and the symmetric boundaries violate the normalization condition. An example is presented in which the results of a complete model with the two-phase kernel are different from those of its one-fourth model. In contrast, the compensated two-phase kernel can generate similar solutions for both the complete and one-fourth models. However, in general, none of the nonlocal kernels can overcome this issue. Therefore, the symmetrical models are not recommended for nonlocal integral elasticity based phase field simulations of MPTs. The current study helps for a better study of nonlocal elasticity based phase field problems for various phenomena such as various PTs.

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as $$1/\epsilon $$ 1 / ϵ , and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$ ϵ → 0 . We establish a $$\Gamma $$ Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $$\Gamma $$ Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

In this article, a two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional (2D) nonlinear coupled wave equations (NCWEqs). In comparison with the existent compact ADI methods for NCWEqs, there are two obvious characters. (1) Numerical solutions at time level one, which have effect on the global stability and convergence of the numerical solutions, are not needed to be solved firstly by using another numerical scheme. (2) The computational cost is comparatively low and acceptable because the new compact ADI method is a linear scheme, thus avoiding Newton’s iterations. By using the energy analysis method, it is shown that numerical solutions converge to exact solutions with an order of ${\mathscr{O}}({{{\varDelta }}} t^{2}+{h^{4}_{x}}+{h^{4}_{y}})$ in H1- and $L^{\infty }$ -norms, and are uniquely solvable. Numerical examples are given to illustrate the exactness of the theoretical findings and the high-performance of our methods.

The purpose of this paper is to study the stability analysis of the curved system made of piezoelectric materials under low-velocity impact. To examine the contact force through the impactor and structure, the Hertz contact model has been taken into account. The minimum potential energy method is applied for achieving the structure’s boundary and governing equations subject to a low-speed impact. In the numerical investigation’s step, the Galerkin approach could be considered as a class of techniques to changing an operator of a continuous problem to a discrete one in the displacement domains. Newmark method is presented for solving the problem in the time domains. After this, by coupling these two approaches (Galerkin and Newmark), the electrically curved system’s governing equations have been solved subjected to external loads for achieving the responses of low-velocity impact and dynamics. The results show that some geometrical and physical factors contribute considerably to the dynamic stability information of the electrically curved system under low-speed impact. The results show that the convergence study of the Galerkin method appears when the circumferential and axial mode numbers are more than 8. This study’s outcomes are likely to be employed can be used in the aircraft industries.

This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

We consider the dynamics of infinite many-dimensional harmonic crystals and study the Cauchy problem with random initial data. Under some restrictions on the interaction between particles of the crystal and distribution of the initial data, we prove the convergence of space-time statistical solutions to a Gaussian measure.

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., $$\mathop {\lim}\limits_{t \to T} A(t) = 0$$ , or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash--Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces $H_*^m$ for $m$ large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable $\varepsilon$--regularization and passing to the limit $\varepsilon\to 0$., Comment: To appear in: Mathematische Annalen

We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined., 40 pages, 3 figures

We present projection-iterative schemes for the realization of the variational-grid (finite-element and local-variation) methods aimed at the solution of various variational problems of mechanics for elastic and elastoplastic solids with inclusions and holes. By using the solutions of the problems of elastoplastic deformation of inhomogeneous isotropic plates and cylindrical shells with holes of various kinds as examples, we analyze the mutual influence of the holes and inclusions. We investigate the problem of convergence of the proposed schemes and demonstrate their efficiency from the viewpoint of reduction of the computer time required for their practical applications.

In this paper we consider a family of three-dimensional problems in thermoelasticity for elliptic membrane shells and study the asymptotic behaviour of the solution when the thickness tends to zero. We fully characterize with strong convergence results the limit as the unique solution of a two-dimensional problem, where the reference domain is the common middle surface of the family of three-dimensional shells. The problems are dynamic and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Fil: Cao Rial, M. T.. Universidade da Coruña; España Fil: Castiñeira, G.. Universidad de Vigo; España Fil: Rodríguez Arós, Á.. Universidade da Coruña; España Fil: Roscani, Sabrina Dina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina

We consider a fractal equation of the integral form with Bessel fractional integrodifferential operator and a positive parameter. In a part of the initial hyperplane, where the limit value has good properties, we establish the property of local strengthening of the convergence of regular solutions with generalized limit values.

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0

In this paper, a h-adaptive methodology based on the polytopal meshes is proposed for capturing high stress gradients at the materials corners and the stress singularities at the vicinity of a crack tip. The adaptive refinement is based on the error indicator directly computed from the displacement solutions of the scaled boundary finite element method. Based on the error indicator, a polygon of n-sides which has an error exceeding a specified tolerance is subdivided recursively into $$(n+1)$$ child polygons. The salient features of the proposed framework are: (a) circumvents a need for post-processing techniques for error estimation; (b) elements with hanging nodes are treated as polygons without a need for special treatment and (c) stress gradients and stress singularities are accurately captured due to the semi-analytical formulation. The robustness and the convergence properties of the proposed framework is demonstrated with three benchmark examples.

We consider a class of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn–Hilliard equation is of viscous type and of pure type.

The effective properties of the fiber-reinforced composite materials with fibers of circle cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For an analytical solution of the periodically repeated cell problem the Schwarz alternating process (SAP) was employed. Convergence of this method was proved by S. Mikhlin, S. Sobolev, V. Mityushev. Unfortunately, the rate of the convergence is often slow, especially for nondilute high-contrast composite materials. For improving this drawback we used Padé approximations for various forms of SAP solutions with the following additive matching of obtained expressions. As a result, the solutions in our paper are obtained in a fairly simple and convenient form. They can be used even for a volume fraction of inclusion very near the physically possible maximum value as well as for high-contrast composite constituents. The results are confirmed by comparison with known numerical and asymptotic results.

The convergence-confinement method is widely used in conventional tunneling at a preliminary stage of the support design. A circular tunnel through the ground in an initially isotropic stress state and the behavior of ground-support interaction simplified utilizing a two-dimensional plane-strain are postulated. From a point of view of practical application, the direct algorithm process so-called the direct calculation method (DCM) is proposed in this paper to deal with solving the solution of stresses/displacements between the ground reaction curve (GRC) and the support confining curve (SCC) in the final equilibrium state by applying the simultaneous equations in the elastic region and using the numerical analysis known as the Newton recursion method for finding roots of the non-linear equations in the plastic region. This explicit procedure also can realize the analytical solution to an executable computation that can be stepwise estimated by using a simple calculation spreadsheet. The validity of the developed method for the analytical solution was examined by the finite element analysis (FEM) to investigate the influence of mechanical properties of the ground and the time-dependent effects of the shotcrete lining-support on the GRC, SCC and stress path at the intrados of the tunnel. The agreement of results between DCM and FEM was found to be excellent in the elastic and elastic-perfectly plastic media.

We consider the problem of diffraction of an acoustic wave on a set of small obstacles (cavities) with the boundary condition of the third kind on the obstacle boundary and the radiation condition at infinity. It is assumed that the obstacle diameters converge and for each obstacle the distance to the nearest obstacle is much greater than the obstacle diameter. We establish the closeness between the solutions to the boundary value problems in a domain with obstacles and the solution to the homogenized problem and prove the convergence of scattering frequencies to the scattering frequencies of the limiting problem with potential.

Dynamic stiffness formulation is proposed in this paper for both transverse and in-plane vibration of rectangular plates that account for arbitrary boundary conditions. A generalized superposition method is developed to obtain the homogeneous solutions for the governing equations of both transverse and in-plane vibration. Consequently, the dynamic stiffness matrices are formed in a more straightforward way by projection method, the dimensions of which are greatly reduced in comparison with those from the conventional Gorman’s superposition method. The finite element technique is utilized to assemble local stiffness matrix into global coordinates so as to address the dynamics of plate assemblies. Various types of plate-like structures are investigated by the proposed method, through which excellent agreement is found between our results and those from finite element method. The effectiveness, accuracy and convergence of the proposed DSM for both transverse and in-plane vibration are proved in several numerical examples, which demonstrates the proposed DSM is an excellent alternative to the existing DSM.

In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings., 15 pages, one figure; To appear in Monatshefte f\"ur Mathematik

This paper focuses on stochastic systems of weakly interacting particles whose dynamics depend on the empirical measures of the whole populations. The drift and diffusion coefficients of the dynamical systems are assumed to be locally Lipschitz continuous and satisfy global linear growth condition. The limits of such systems as the number of particles tends to infinity are studied, and the rate of convergence of the sequences of empirical measures to their limits in terms of pth Monge-Wasserstein distance is established. We also investigate the existence, uniqueness, and boundedness, and continuity of solutions of the limiting McKean-Vlasov equations associated to the systems.

We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

This paper aims to present the analysis of transient wave characteristics in a functionally graded viscothermoelastic infinite medium with spherical cavity in the context of generalized thermoelasticity. Continued series solution is used to solve simultanious differential equations for evaluating the field variables. Convergence of the series solution is implemented and investigated that the series of functions are absolutely and uniformly convergent. The formal solution for the field variables are obtained analytically and represented graphically. The effect of grading index and different theories of generalized thermoelasticity are also shown graphically to examine the behavior of the variations of the field variables.

In this survey paper we discuss a series of recent results concerning nonnegative solutions to nonlinear diffusion equations of porous-medium type on Cartan–Hadamard manifolds, a special class of negatively-curved Riemannian manifolds that generalize the Euclidean space. We focus on sharp barrier estimates, asymptotic convergence and smoothing effects, describing quantitatively how the curvature behavior at infinity affects the way solutions depart from having a Euclidean-like structure.

We develop and analyze an ultraweak variational formulation of the Reissner–Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness t. We also prove weak convergence of the Reissner–Mindlin solution to the solution of the corresponding Kirchhoff–Love model when $$t\rightarrow 0$$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov–Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory covers the case of non-convex polygonal plates. A numerical experiment for some smooth model solutions with fixed load confirms that our scheme is locking free.

This paper deals with analyzing the nonlinear vibration of an isotropic cracked plate interacting with an air cavity. A part-through surface crack with variable orientations and positions is considered and modeled using the modified line spring model. In the first step, based on the Von Karman theory, the governing equation of the nonlinear vibration related to the cracked plate–cavity is presented. Then, by employing the Euler equation along with the Galerkin method, the coupling effect between the fluid–solid media inside the enclosure is eliminated. In the next step, the variational iteration method (VIM) is introduced as an appropriate method for nonlinear analysis of the mentioned system. To this end, the convergence of the nonlinear coupled natural frequencies with high precision is proved by performing four iterations of VIM. Finally, the effect of the length, angle, and position corresponding to the crack as well as the cavity depth on the frequency ratio is inspected for various boundary conditions by plotting three and four-dimensional backbone curves. It is revealed that the crack angle is the most effective parameter on the frequency ratio.

The classical problem of the construction of C 1 conforming single-patch quadrilateral finite elements has been solved in this investigation by using the blending function interpolation method. In order to achieve the C 1 conformity on the interfaces of quadrilateral elements, complete second-order derivatives are used at the element vertices, and the information of geometrical mapping is also considered into the construction of shape functions. It is found that the shape functions and the polynomial spaces of the present elements vary with element shapes. However, the developed quadrilateral elements are at least third order for general quadrilateral shapes and fifth order for rectangular shapes. Therefore, very fast convergence can be achieved. A promising feature of the present elements is that they can be used in cooperation with those high-precision rectangular and triangular elements. Since the present elements are over conforming on element vertices, an approach for handling problems of material discontinuity is also proposed. Numerical examples of Kirchhoff plates are employed to demonstrate the computational performance of the present elements.

In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model. Firstly, the time fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short-pulse laser heating is built. Laplace and Fourier cosine transforms are performed to derive the semi-analytical expression of temperature distribution in the Laplace domain. Then, by the L1 approximation for the Caputo derivative, the finite difference algorithm is developed for the short-pulse laser heating problem. The solvability, stability, and convergence of this algorithm are also examined. Meanwhile, the efficiency and accuracy of this method have been verified by using three numerical examples. Finally, based on numerical analysis, we study the non-Fourier heat conduction behavior and discuss the effect of variability of parameters, such as fractional parameter and the ratio between the relaxation and retardation times, on the temperature distribution graphically. We believe that this analysis, besides benefiting the laser heating applications, will also provide a deep theoretical insight to interpret the anomalous heat transport process.

We propose a fast method for high order approximations of the solution to the Cauchy problem for the linear nonstationary Stokes system in ℝ3 in the unknown velocity u and kinematic pressure P. The density f (x, t) and the divergence-free vector initial value g(x) are smooth and rapidly decreasing as |x| → ∞. We construct the vector u = u1+u2 where u1 solves a system of homogeneous heat equations and u2 solves a system of nonhomogeneous heat equations with right-hand side f −∇P, where P = − (∇ · f) and denotes the harmonic potential. Fast semianalytic cubature formulas for computing the harmonic potential and the solution to the heat equation based on the approximation of the data by functions with analitically known potentials are considered. The gradient ∇P can be approximated by the gradient of the cubature of P, which is a semianalytic formula. We derive fast and accurate high order formulas for approximation of u1, u2, P, and ∇P. The accuracy of the method and the convergence order 2, 4, 6, 8 are confirmed by numerical experiments.

The present investigation is focused on the buckling behavior of strain gradient nonlocal beam embedded in Winkler elastic foundation. The first-order strain gradient model has been combined with the Euler–Bernoulli beam theory to formulate the proposed model using Hamilton’s principle. Three numerically efficient methods, namely Haar wavelet method (HWM), higher order Haar wavelet method (HOHWM), and differential quadrature method (DQM) are employed to analyze the buckling characteristics of the strain gradient nonlocal beam. The impacts of several parameters such as nonlocal parameter, strain gradient parameter, and Winkler modulus parameter on critical buckling loads are studied effectively. The basic ideas of the numerical methods, viz. HWM, HOHWM, and DQM are presented comprehensively. Also, a comparative study has been conducted to explore the effectiveness and applicability of all the three numerical methods in terms of convergence study. Finally, the results, obtained by this investigation, are validated properly with other works published earlier.

We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method. ISSN:1615-3375 ISSN:1615-3383

The Navier–Stokes–Maxwell–Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.

In this paper, a Lie-group integrator based on $$GL_4(\mathbb {R})$$ and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.

A systematic numerical integration method is applied to the absolute nodal coordinate formulation (ANCF) fully parameterized beam element with smooth varying and continuous cross section. Moreover, the formulation for the integration points and weight coefficients are given in the method which is used to model the multilayer beam with a circular cross section. To negate the effect of the bending stiffness for the element used to model the high-voltage electrical wire, the general continuum mechanical approach is adjusted. Additionally, the insulation cover for some particular types of the wire is described by the nearly incompressible Mooney–Rivlin material model. Finally, a static problem is presented to prove the accuracy and convergence properties of the element, and a dynamic problem of a flexible pendulum is simulated whereby the balance of the energy can be ensured. An experiment is carried out in which a wire is released as a pendulum and falls on a steel rod. The configurations of the wire are captured by a high-speed camera and compared with the simulation results. The feasibility of the wire model can therefore be demonstrated.

We study the absolute continuity and local limit theorems for homogeneous functionals defined on configurations of point processes (p.p.s). For empirical p.p.s, we show that under mild hypotheses the distribution of such a functional has a density. Moreover, we present results on convergence in total variation of this distribution to some limit.

In this paper, we study the low Mach number limit of the full compressible Navier–Stokes equations with revised Maxwell law in $$\mathbb {R}^3$$ . By applying the uniform estimates of the error system, we prove that the solutions of the full compressible Navier–Stokes equations with time relaxation converge to that of the incompressible Navier–Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

In our book “Navier–Stokes Equations in Planar Domains”, Imperial College Press, 2013, we have suggested a fourth-order compact scheme for the Navier–Stokes equations in streamfunction formulation $$\partial _t(\Delta \psi )+(\nabla ^{\perp }\psi ) \cdot \nabla (\Delta \psi ) =\nu \Delta ^2 \psi $$ . Here we present a new approach for the analysis of a high-order compact scheme for the Navier–Stokes equations in cases where the convective term vanishes, or in cases where the viscous term dominates the convective term. In these cases the Navier–Stokes equations is replaced by the time-dependent Stokes equation $$\partial _t(\Delta \psi )=\nu \Delta ^2 \psi $$ . The same type of fourth-order compact schemes that were proposed for the Navier–Stokes equations, may be adopted to the time-dependent Stokes problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence is actually four, thus the error tends to zero as $$O(h^4)$$ , where h is the size of the mesh.

The Cauchy problem of a generalized Poisson–Nernst–Planck–Navier–Stokes system in $${\mathbb {R}}^3$$ will be considered in this article. Based on the spectral analysis and the energy method, under some assumptions of the initial data, we obtain the lower bound and upper bound decay rates of the solution, which shows that the solution will converge to its constant equilibrium state at the same $$L^2$$ -decay rates as the linearized one and the convergence rates are optimal.

In this paper we will analyze the convergence of the non-overlapping double sweep domain decomposition method (DDM) with transmission conditions based on PMLs for the Helmholtz equation. The main goal is to establish the convergence of the double sweep DDM of both the continuous level problem and the corresponding finite element problem. We show that the double sweep process can be viewed as a contraction mapping of boundary data used for local subdomain problems not only in the continuous level and but also in the discrete level. It turns out that the contraction factor of the contraction mapping of the continuous level problem is given by an exponentially small factor determined by PML strength and PML width, whereas the counterpart of the discrete level problem is governed by the dominant term between the contraction factor similar to that of the continuous level problem and the maximal discrete reflection coefficient resulting from fast decaying evanescent modes. Based on this analysis we prove the convergence of approximate solutions in the $$H^1$$ -norm. We also analyze how the discrete double sweep DDM depends on the number of subdomains and the PML parameters as the finite element discretization resolves sufficiently the Helmholtz and PML equations. Our theoretical results suggest that the contraction factor for the propagating modes depends linearly on the number of subdomains. To ensure the convergence, it is sufficient to have the PML width growing logarithmically with the number of subdomains. In the end, numerical experiments illustrating the convergence will be presented as well.

In this paper, we develop and study the mixed finite element method for a hemivariational inequality of the stationary Navier–Stokes equations (NS hemivariational inequality). The NS hemivariational inequality models the motion of a viscous incompressible fluid in a bounded domain, subject to a nonsmooth and nonconvex slip boundary condition. The incompressibility contraint is treated through a mixed formulation. Solution existence and uniqueness are explored. The mixed finite element method is applied to solve the NS hemivariational inequality and error estimates are derived. Numerical results are reported on the use of the P1b/P1 pair, illustrating the optimal convergence order predicted by the error analysis.

The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. The cost of computing the polarizability operator generally scales as $${\mathcal {O}}(N_{e}^4)$$ where $$N_e$$ is the number of electrons in the system. The recently developed adaptively compressed polarizability operator (ACP) formulation [L. Lin, Z. Xu and L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to $${\mathcal {O}}(N_{e}^3)$$ in the context of phonon calculations with a large basis set for the first time, and demonstrates its effectiveness for model problems. In this paper, we improve the performance of the ACP formulation by splitting the polarizability into a near singular component that is statically compressed, and a smooth component that is adaptively compressed. The new split representation maintains the $${\mathcal {O}}(N_e^3)$$ complexity, and accelerates nearly all components of the ACP formulation, including Chebyshev interpolation of energy levels, iterative solution of Sternheimer equations, and convergence of the Dyson equations. For simulation of real materials, we discuss how to incorporate nonlocal pseudopotentials and finite temperature effects. We demonstrate the effectiveness of our method using one-dimensional model problem in insulating and metallic regimes, as well as its accuracy for real molecules and solids.

A new finite element method is proposed for second order elliptic interface problems based on a local anisotropic fitting mixed mesh. The local anisotropic fitting mixed mesh is generated from an interface-unfitted mesh by simply connecting the intersected points of the interface and the underlying mesh successively. Optimal approximation capabilities on anisotropic elements are proved, the convergence rates are linear and quadratic in $$H^1$$ and $$L^2$$ norms, respectively. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a new multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results.

In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.