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The objective of this paper is to discuss finite-time bounded synchronization for a class of the growing complex network with nondelayed and delayed coupling. In order to realize finite-time synchronization of complex networks, a new finite-time stable theory is proposed; effective criteria are developed to realize synchronization of the growing complex dynamical network in finite time. Moreover, the error of two growing networks is bounded simultaneously in the process of finite-time synchronization. Finally, some numerical examples are provided to verify the theoretical results established in this paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present an incremental pressure correction finite element method for the time-dependent Oldroyd flows. This method is a fully discrete projection method. As we all know, most projection methods have been studied without space discretization. Then the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on a time discrete error and a spatial discrete error. In order to show the effectiveness of the method, we also present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time-dependent Oldroyd flows. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [ABSTRACT FROM AUTHOR]

Highlights • A penalty finite element method for the steady MHD equations was given. • The solution of the penalty method convergence the solution of the steady MHD equations. • The stability analysis shows our method is stable. • The error estimate shows our method has an optimal convergence order. • The numerical results of the Hartmann flow was shown. Abstract In this paper, we give a penalty finite element method for the steady MHD equations. In this method, we decouple the MHD into two equations, one for the velocity and magnetic (u , B) , the other for the pressure p. We prove the existence of the penalty method and the optimal error estimate. Then, we give the penalty finite element method for the MHD equations. The stability analysis shows our method is stable, and the error estimate shows our method has an optimal convergence order. Finally, we give some numerical results of exact solution equation and Hartmann flow. The numerical results show that our penalty finite element method is effect. [ABSTRACT FROM AUTHOR]

We study in this paper the filament-based lamellipodium model (FBLM) and the corresponding finite element method (FEM) used to solve it. We investigate fundamental numerical properties of the FEM and justify its further use with the FBLM. We show that the FEM satisfies a time step stability condition that is consistent with the nature of the problem and propose a particular strategy to automatically adapt the time step of the method. We show that the FEM converges with respect to the (two-dimensional) space discretization in a series of characteristic and representative chemotaxis and haptotaxis experiments. We embed and couple the FBLM with a complex and adaptive extracellular environment comprised of chemical and adhesion components that are described by their macroscopic density and study their combined time evolution. With this combination, we study the sensitivity of the FBLM on several of its controlling parameters and discuss their influence in the dynamics of the model and its future evolution. We finally perform a number of numerical experiments that reproduce biological cases and compare the results with the ones reported in the literature. [ABSTRACT FROM AUTHOR]

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1) in the H1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1 + τ2) in the H1 norm can be obtained in the Crank-Nicolson fully discrete scheme. [ABSTRACT FROM AUTHOR]

We consider a vector-Laplace problem posed on a two-dimensional surface embedded in a three-dimensional domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included. [ABSTRACT FROM AUTHOR]

If an elliptic differential operator associated with an H(curl)-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H(curl)-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest-order Nédélec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H(curl)-stable, and quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh size) in the H(curl) norm are obtained provided the right-hand side belongs to H(div). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first-order corrector, including corresponding quantitative error estimates without the requirement of scale separation. The constructed generalized finite element method falls into the class of localized orthogonal decomposition methods, which have not been studied for H(curl)-problems so far. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Kármán equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables. Numerical results that justify the theoretical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, a new approach of superconvergent estimate of bilinear finite element is established for two-dimensional time-fractional diffusion equation under fully-discrete scheme. The novelty of this approach is the combination technique of the interpolation and Ritz projection as well as the superclose estimate in H 1 -norm between them, which avoids the difficulty of constructing a postprocessing operator for Ritz projection operator, and reduces the regularity requirement of the exact solution. At the same time, three numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics. [ABSTRACT FROM AUTHOR]

We present new results where free parameters in spurious oscillations at layers diminishing (SOLD) method are adaptively chosen. Provided numerical results are for conforming piecewise linear finite element space with free parameters from piecewise constant finite element space. We focus on a higher order convergence we discovered in previous paper by Lukáš (2015). [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative approach to a priori $$L^\infty $$ -error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all. [ABSTRACT FROM AUTHOR]

Abstract: Spectral elements methods provide very accurate solutions of elliptic problems. In this paper we apply the method to the A N (i.e. SP2N−1) approximation of neutron transport. Numerical results for classical benchmark cases highlight its performance in comparison with finite element computations, in terms of accuracy per degree of freedom and convergence rate. All calculations presented in this paper refer to two-dimensional problems. The method can easily be extended to three-dimensional cases. The results illustrate promising features of the method for more complex transport problems. [Copyright &y& Elsevier]

This paper presents numerical procedures for the treatment of tensors related to the vector Jiles–Atherton (JA) model applied to the finite element method (FEM). The method is based on the 3-D differential permeability formulation. A more precise calculation implies the determination of the off-diagonal terms of tensor parameters. The main goal is to verify if such a change improves the convergence stability of the model coupled with FEM and if the use of the JA off-diagonal terms yields a more accurate hysteresis representation. We present results comparing experimental data of the Testing Electromagnetic Analysis Methods problem 32 with the proposed method. [ABSTRACT FROM AUTHOR]

A new highly efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations is studied in this paper. The algorithm uses a lower order finite element pair (i.e., P 1 b − P 1 − P 1 ) to compute an initial approximation, that is using the Mini-element (i.e., P 1 b − P 1 ) to approximate the velocity and pressure and P 1 element to approximate the magnetic field, then applies a higher order finite element pair (i.e., P 2 − P 1 − P 2 ) to solve a linear system on the same mesh. Furthermore, the convergence analyses of standard Galerkin finite element method and the two-step algorithm are addressed. Lastly, numerical experiments are presented to verify both the theory and the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

The finite-element method (FEM) has already been extended to analyze transport properties of electron waves of two-dimensional electron systems in magnetic fields. Although many researchers have created new formulations or improvements to this method, few have analyzed how this method is applied to realistic systems. The present paper suggests that conventional formulations of the FEM do not give accurate results for large systems or for strong magnetic fields; in addition, it suggests that the selected gauge significantly influences the numerical results. Furthermore, this paper proposes a conceptually different formulation of the FEM that solves the poor convergence problem. This formulation is simple: matrix elements are multiplied by the Peierls phase in the absence of a magnetic field. To show the advantages of this formulation, numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, a new mixed variational form for the Reissner-Mindlin problem is given, which contains two unknowns instead of the classical three ones. A mixed triangle finite element scheme is constructed to get a discrete solution. A new method is put to use for proving the uniqueness of the solutions in both continuous and discrete mixed variational formulations. The convergence and error estimations are obtained with the help of different norms. Numerical experiments are given to verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

A concave finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is presented for scattering by a large body with nonuniform deep cavities. Different from the conventional FE-BI-MLFMA, the boundary integral equation in this concave FE-BI-MLFMA is established on a concave surface to reduce the region of the finite element method (FEM), which can significantly reduce the dispersion error from the FEM and improve the efficiency of FE-BI-MLFMA especially for nonuniform cavities. To eliminate the problem of slow convergence caused by concave surface, an efficient preconditioner based on the sparse approximate inverse (SAI) is constructed in this paper. Numerical performance of the constructed preconditioner based on the SAI is investigated in detail. Numerical experiments demonstrate the accuracy and efficiency of this SAI preconditioned concave FE-BI-MLFMA for nonuniform deep and large cavites. This SAI preconditioned concave FE-BI-MLFMA is parallelized to further improve its capability in this paper. An extremely big and deep nonuniform cavity has been calculated, demonstrating the great capability of this parallel concave FE-BI-MLFMA. [ABSTRACT FROM PUBLISHER]

This paper presents a deflation technique to improve the convergence of finite element (FE) analyses with infinite elements. In FE analyses of electromagnetic fields, large air regions must be discretized into FE meshes. This leads to increases in computational time. The infinite element in which electromagnetic fields in air region are accurately expressed has been introduced in order to solve this problem. However, when using the infinite element, convergence of iterative liner solvers deteriorates because the condition number of FE matrices becomes large. In this paper, a deflation technique to improve convergence of iterative solvers is introduced. Numerical examples show that the proposed technique can improve convergence characteristics in a magnetostatic analysis with finite and infinite elements. [ABSTRACT FROM PUBLISHER]

Abstract: This paper presents a novel three-dimensional finite element for gradient elasticity. The new finite element BR153L9 is a straightforward extension of the two-dimensional element QU34L4 developed by Shu et al. (1999) , which is based on a mixed-type formulation. Within this paper the derivation of the mixed-type finite element scheme is reviewed and details of the implementation are given. Finally, numerical results of an extended patch test and a benchmark test with the three-dimensional finite element are presented in order to validate the formulation and prove the convergence. [Copyright &y& Elsevier]

Abstract: The yield vertex non-coaxial model is different from classical elastoplastic models, in that there is an additional plastic strain rate tangential to yield surfaces, as well as the plastic strain rate normal to yield surfaces, when orientations of principal stress change. This feature raises concerns on its finite element implementations. In nonlinear finite element numerical iterations, a large tangential plastic strain rate is likely to make the trial total strain rate direct inside a yield surface, which entails convergence difficulty. Some modifications are introduced on the non-coaxial model itself to make numerical convergence easier in the work published in Yang and Yu (2010) . This paper is an extension of the previous work. Instead of modifying the non-coaxial model itself, this paper concerns the use of finite element explicit procedure, which is suitable for highly discontinuous problems. The simulations of shallow foundation load-settlement responses indicate that the finite element explicit procedure, assisted with a robust and explicit automatic substepping integration scheme of the non-coaxial model, does not encounter numerical difficulty. In addition, the overall trends of implicit and explicit simulations are similar. [Copyright &y& Elsevier]

Abstract: The Finite Element Method is the most commonly used prediction technique to model the harmonic behavior of poroelastic materials. A major disadvantage of this method is its practical frequency limitation in that the computational efforts become prohibitively large at higher frequencies. A recently developed Wave Based Method is an efficient alternative deterministic prediction technique which aims at relaxing this frequency limitation by using exact solutions of the governing equations to approximate the field variables. This paper discusses the application of the Wave Based Method for the particular case that stress singularities are present in corners of the poroelastic domain. Based on an asymptotic analysis, the paper derives a criterion to predict the presence of stress singularities and proposes a suitable set of enrichment functions to extend the conventional set of expansion functions. The beneficial effect of incorporating these functions on the convergence of the Wave Based Method is illustrated by means of a numerical validation study. [Copyright &y& Elsevier]

Abstract: The paper proposes Luenberger-like observers for a rotating body-beam system. The latter is described by a partial differential equation and its dynamic is governed a unitary group. The observers proposed in the paper are able to reconstitute the dynamic evolution of the beam profile, by measuring the moment force on the boundary only. Meanwhile we propose a reliable numerical scheme, based on a finite element method, in order to simulate the observation system and the observers. The efficiency of our proposed scheme and the performances of the observers are illustrated with numerical simulation results. [ABSTRACT FROM AUTHOR]

Abstract: The main aim of the paper is to provide the generalized stochastic perturbation technique based on the classical Taylor expansion with a single random variable. The main problem discussed below is an application of this expansion to the solution of various partial differential equations with random coefficients by the fundamental numerical methods, i.e. Boundary Element Method, Finite Element Method as well as the Finite Difference Method. Since th order expansion is employed for this purpose, the probabilistic moments of the solution can be determined with a priori given accuracy. Contrary to the second order techniques used before, a perturbation parameter is also included in the relevant approximations, so that the overall solution convergence can be sped up by some modification of its value. Application of computational methodologies presented in transient problems (dynamics or heat transfer) are also commented on in the paper, together with stochastic processes modelling by the double Taylor expansion. [Copyright &y& Elsevier]

A general superconvergence result is established for the stabilized finite element approximations for the stationary Stokes equations. The superconvergence is obtained by applying the L² projection method for the finite element approximations and/or their close relatives. For the standard Galerkin method, existing results show that superconvergence is possible by projecting directly the finite element approximations onto properly defined finite element spaces associated with a mesh with different scales. But for the stabilized finite element method, the authors had to apply the L² projection on a trivially modified version of the finite element solution. This papers shows how the modification should be made and why the L² projection on the modified solution has superconvergence. Although the method is demonstrated for one class of stabilized finite element methods, it can certainly be extended to other type of stabilized schemes without any difficulty. Like other results in the family of L² projection methods, the superconvergence presented in this paper is based on some regularity assumption for the Stokes problem and is valid for general stabilized finite element method with regular but non-uniform partitions. [ABSTRACT FROM AUTHOR]

Our previous paper proposed two frameworks for iterative linear solvers: the implicit and explicit error correction methods. In this paper, we discuss the convergence property of these methods. A formula we derive explains the reasonability of the auxiliary matrix that Kameari suggested for thin elements. Additionally, an enhanced auxiliary matrix is devised for thin elements, in which the material property changes discontinuously. [ABSTRACT FROM AUTHOR]

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted L2 projection onto H(div) conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps. [ABSTRACT FROM AUTHOR]

The main issue in the present paper is to comment and illustrate on new acoustic examples the method of analysis of modal series, termed "method of orthocomplement", that has been recently proposed by the authors to improve the convergence of such series. The general method consists in a direct analysis and transformation of the remainders of ordinary series. It results in a family of "hybrid" modal representations involving an ordinary modal sum of order N, a "quasi-static" term based on the N first modes, and an "accelerated" modal series. Using the transformed modal formulae eliminates the Gibbs oscillations - that are attached in infinite dimensional models to modal boundary discontinuities - and also the consequences of such phenomena on finite element approximations. The method is applied in the present paper to plane waves in acoustic tubes and to 3D acoustic fields inside a car compartment, in view of the synthesis of acoustic receptances or impedances to be used in practical acoustic design. The main technical difficulty being the treatment of singular linear boundary problems or systems of linear equations that arise during the study of closed rigid cavities or tubes, a whole section of the paper had thus to be devoted to pseudo-inversion. [ABSTRACT FROM AUTHOR]

We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poisson's equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported. [ABSTRACT FROM AUTHOR]

This paper is devoted to the rigorous obtention of the energy balance in porous materials. The wave propagation in the porous media is described by Biot-Allard’s {u,U} and {u,P} formulations. The paper derives the expressions for stored kinetic and strain energies together with dissipated energies. It is shown that, in the case of mixed formulations, these expressions do not correspond to the real and imaginary parts of the variational formulations. A quantitative convergence analysis of finite element scheme is then undertaken with the help of these indicators. It is shown that the order of convergence of these indicators for linear finite-element is one and that they are then well fitted to check the validity of finite-element models. [ABSTRACT FROM AUTHOR]

This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz­Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers. [ABSTRACT FROM AUTHOR]

In this paper we apply the generalized Taylor­Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rig- orously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic dis- continuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also con- sidered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver. [ABSTRACT FROM AUTHOR]

In this paper, a new mixed finite element scheme is proposed for the nonlinear Sobolev equation by employing the finite element pair Q 11 / Q 01 × Q 10 . Based on the combination of interpolation and projection skill as well as the mean-value technique, the τ -independent superclose results of the original variable u in H 1 -norm and the variable q → = − ( a ( u ) ∇ u t + b ( u ) ∇ u ) in L 2 -norm are deduced for the semi-discrete and linearized fully-discrete systems ( τ is the temporal partition parameter). What’s more, the new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained unconditionally, while previous literature always require certain time step conditions. Finally, some numerical results are provided to confirm our theoretical analysis, and show the efficiency of the method. [ABSTRACT FROM AUTHOR]

We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric constant ε r ( x ) , x ∈ R 3 , which is an unknown coefficient in Maxwell’s equations. On the first stage the globally convergent method of Beilina and Klibanov (2012) is applied to get a good first approximation for the exact solution. Results of this stage were presented in Thành et al. (2014). On the second stage the locally convergent adaptive finite element method of Beilina (2011) is applied to refine the solution obtained on the first stage. The two-stage numerical procedure results in accurate imaging of all three components of interest of targets: shapes, locations and refractive indices. In this paper we briefly describe methods and present new reconstruction results for both stages. [ABSTRACT FROM AUTHOR]

Analytical solutions for tunnel design are widely used in practical engineering, as they allow a quick analysis of design issues such as estimation of support requirement. In recent years, several papers analyzing the behavior of rock masses that obey the conventional or generalized Hoek–Brown criterion have been published. This article presents a complementary analysis that includes a new normalization of the generalized Hoek–Brown failure criterion, complete solutions for associated and non-associated flow rules, with some new closed-form solutions in the latter case, and in-depth considerations regarding intermediate stresses and edge effects. The results obtained show full agreement with existing solutions in the literature, when possible, and with numerical finite element models in cases that had not been treated previously. [ABSTRACT FROM AUTHOR]

In this paper, we generalize one first order nonconforming quadrilateral finite element proposed by Lin, Tobiska and Zhou to any odd order. We construct degrees of freedom for shape function spaces for this family of elements and show their unisolvency. In addition, we present a medium a priori error analysis for this family of nonconforming elements on general quadrilateral meshes. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H 1 regularity of the exact solution. Numerics are presented to demonstrate theoretical results which in particular show dependence of convergence on mesh distortion parameters α . [ABSTRACT FROM AUTHOR]

In this paper, a new gradient projection method is proposed, which generates a feasible matrix sequences. The decent property of this method is proved. Based on the decent property, the convergence of the new method is discussed. Moreover, a sufficient and necessary condition for the optimal matrix is obtained. Finally, numerical experiments show the new method is effective in precision. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Stokes-Darcy flow problem by combining well-known mixed finite elements that are separately convergent for Stokes and Darcy problems. In the approach in which the Darcy problem is set in its natural H(div) formulation and the Stokes problem is expressed in velocity0pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the global velocity space. Our analysis of the Galerkin discretization of the resulting problem reveals that if the mixed finite element space used in the Darcy domain admits an H(div)-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates. [ABSTRACT FROM AUTHOR]

This paper is concerned with the scattering problem of a rectangular cavity. We solve this problem by a least-squares nonpolynomial finite element method. In the method, we use Fourier-Bessel functions to capture the behaviors of the total field around corners. And the scattered field towards infinity is represented by a combination of half-space Green functions. Then we analyze the convergence and give an error estimate of the method. By coupling the least-squares nonpolynomial finite element method and the Newton method, we proposed an algorithm for the inverse scattering problem. Numerical experiments are presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of elliptic interface problems via isoparametric finite element methods. First, a convergence analysis is carried which asserts that optimal rates of convergence are recovered in both the energy and the L 2 –norms. Subsequently, the efficiency of isoparametric finite elements for the problem at hand is also assessed via numerical experiments. Results both with smooth and piecewise regular interfaces are presented and discussed. The numerical predictions corroborate the theoretical results and they also indicate that second-order convergence is achieved in the L ∞ -norm. Comparisons between isoparametric finite elements, Lagrangian finite elements and the Immersed Interface Method are also performed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a new discrete weak gradient operator and a new weak Galerkin (WG) finite element method for second order Poisson equations based on this new operator. This newly defined discrete weak gradient operator allows us to use a single stabilizer which is similar to the one used in the discontinuous Galerkin (DG) methods without having to worry about choosing a sufficiently large parameter. In addition, we will establish the optimal convergence rates and validate the results with numerical examples. [Copyright &y& Elsevier]

In this paper, a numerical model to simulate a multiphase flow interacting with deformable solid bodies is proposed. The fluid domain is modeled through the lattice Boltzmann method and the Shan–Chen model is adopted to handle the multiphase feature. The interaction of the flow with immersed solid bodies is accounted for by using the Immersed Boundary method. Corotational beam finite elements are used to model the deformable bodies and non-linear structure dynamics is predicted through the Time Discontinuous Galerkin method. A numerical campaign is carried out in order to assess the effectiveness and accuracy of the proposed modeling by involving different scenarios. In particular, the model is validated by performing the bubble test and by comparing present results with the ones from a numerical commercial software. Moreover, the properties in terms of convergence are discussed. In addition, the effectiveness of the proposed methodology is evaluated by computing the error in terms of the energy that is artificially introduced in the system at the fluid–solid interface. Present findings show that the proposed approach is robust, accurate and suitable of being applied to a lot of practical applications involving the interaction between multiphase flows and deformable solid bodies. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide a-priori error estimates for problems involving curved interfaces and solved with the linear or quadratic extended finite-element method (X-FEM), with particular emphasis on the influence of the geometry representation and the quadrature. We focus on strong discontinuity problems, which covers the case of holes in a material or cracks not subjected to contact as the main applications. The well-known approximation of the curved geometry based on the interpolated level-set function and straight linear or curved quadratic subcells is used, whose accuracy is quantified by means of an appropriate error measure. A priori error estimates are then derived, which depend upon the interpolation order of the displacement, and foremost upon the above error measure and the quadrature scheme in the subcells. The theoretical predictions are successfully compared with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a two-level defect-correction Oseen iterative finite element method is presented for the stationary conduction–convection equations based on local Gauss integration. The method combines the defect-correction method, the two-level strategy, and the locally stabilized method. The stability and convergence of the proposed method are deduced. Finally, numerical examples verify the theoretical results of the proposed algorithm and show that it is highly efficient and reliable for the considered problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form as where is a Q-Wiener process on the Hilbert space H and where the time kernel b is the locally integrable potential , , or slightly more general. The operator A is unbounded, linear, self-adjoint, and positive on H. Our main assumption concerning the noise term is that is a Hilbert–Schmidt operator on H for some . The numerical approximation is achieved via a standard continuous finite element method in space (parameter h) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter ). We show that for twice continuously differentiable test function with bounded second derivative, for any . This is essentially twice the rate of strong convergence under the same regularity assumption on the noise. [Copyright &y& Elsevier]

This paper aims to develop a fully discrete local discontinuous Galerkin finite element method for numerical simulation of the time-fractional telegraph equation, where the fractional derivative is in the sense of Caputo. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. The stability and convergence of this discontinuous approach are discussed and theoretically proven. Finally numerical examples are performed to illustrate the effectiveness and the accuracy of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new multiscale finite element method for the stationary Navier–Stokes problem. This new method for the lowest equal order finite element pairs is based on the multiscale enrichment and derived from the Navier–Stokes problem itself. Therefore, the new multiscale finite element method better reflects the nature of the nonlinear problem. The well-posedness of this new discrete problem is proved under the standard assumption. Meanwhile, convergence of the optimal order in the -norm for the velocity and the -norm for the pressure is obtained. Especially, via applying a new dual problem and some techniques in the process for proof, we establish the convergence of the optimal order in the -norm for the velocity. Finally, numerical examples confirm our theory analysis and validate the effectiveness of this new method. [Copyright &y& Elsevier]