finite element method

A numerical approximation combining the fast finite difference method in time and the finite element method in space is studied to solve the distributed-order time and Riesz space fractional Schr\"{o}dinger equation. In this work, a fast evaluation of the distributed-order time fractional derivative based on graded time mesh is applied to the time approximation of this equation. Also, the finite element method is used for space approximation. Moreover, the stability and convergence of the resulting discrete scheme are discussed. Finally, some numerical examples are presented to confirm the theoretical results.

Our contribution consists of studying numerical methods based on finite element space and finite difference schema in time of the linear one-dimensional thermoelastic Bresse system with second sound. We establish some a priori error estimates, and present some numerical analysis results of discrete energy under different decay rate profiles. Moreover, we study the behaviors of discrete energy with respect to the system parameters and the initial data. Some numerical simulations will be given in order to validate the theoretical results.

In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand side. To discretize the problem, we utilize a finite element method for the operator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton–Jacobi–Bellman equation. For linearizing the problem on the coarse grid, we employ Newton’s method as an external interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem.

This study delves into the potential polynomial and rational wave solutions of the Kudryashov–Sinelshchikov equation. This equation has multiple applications including the modeling of propagation for nonlinear waves in various physical systems. Through detailed numerical simulations using the finite element approach, we present a set of accurate solitary and soliton solutions for this equation. To validate the effectiveness of our proposed method, we utilize a collocation finite element approach based on quintic B-spline functions. Error norms, including L2 and L∞, are employed to assess the precision of our numerical solutions, ensuring their reliability and accuracy. Visual representations, such as graphs derived from tabulated data, offer valuable insights into the dynamic changes of the equation over time or in response to varying parameters. Furthermore, we compute conservation quantities of motion and investigate the stability of our numerical scheme using Von Neumann theory, providing a comprehensive analysis of the Kudryashov–Sinelshchikov equation and the robustness of our computational approach. The strong alignment between our analytical and numerical results underscores the efficacy of our methodology, which can be extended to tackle more complex nonlinear models with direct relevance to various fields of science and engineering.

The Schrödinger equation with variable-order fractional operator is a challenging problem to be solved numerically. In this study, an implicit fully discrete continuous Galerkin finite element method is developed to tackle this equation while the fractional operator is expressed with a nonsingular Mittag-Leffler kernel. To begin with, the finite difference scheme known as the L1 formula is employed to discretize the temporal term. Next, the continuous Galerkin method is used for spatial discretization. This combination ensures accuracy and stability of the numerical approximation. Our next step is to conduct a stability and error analysis of the proposed scheme. Finally, some numerical results are carried out to validate the theoretical analysis.

The free vibrations of a rod are governed by a differential equation of the form $(a(x)y^\prime)^\prime+\lambda a(x)y(x)=0$, where $a(x)$ is the cross sectional area and $\lambda$ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form $(K-\Lambda M)u=0$ and, for given $a(x)$, we correct the eigenvalues $\Lambda$ of the matrix pair $(K,M)$ to approximate the eigenvalues of the rod equation. The results show that with step size $h$ the correction technique reduces the error from $O(h^2i^4)$ to $O(h^2i^2)$ for the $i$-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient $a(x)$ from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms.

Given the brittle behavior and low flexibility of unreinforced masonry walls, using flexible fiber-reinforced concretes, such as engineered cementitious composites (ECCs), to retrofit them is of great importance. In this regard, the performance of these materials (as a reinforcement layer of unreinforced masonry walls) in improving their behavior (flexibility and strength) against dynamic loads, especially impact loads, should be taken into account. The present study evaluates and compares the vertical middle displacement, energy, and distribution of plastic strains in unreinforced masonry materials, under two conditions, i.e., non-retrofitted and retrofitted with a one-side cover (in the lower surface of the specimen) or two-side cover of ECC layers under dynamic impact loading, through nonlinear dynamic impact analysis. The ECC reinforcement layers in two-side covers were investigated by changing their location (in the middle or edges of the lower surfaces of the specimen), connection type (disconnection or full connection) of the one-side cover of the bottom of the specimen with the ECC reinforcement layer and elastic modulus (from 15 GPa to 22.5 GPa). The retrofitting of the unreinforced masonry materials with ECC layers (especially with two-side cover) improved their behavior against out-plane impact loads, dissipated energy, and reduced the plastic strains and cracks.

Despite the widespread and successful use of FRP sheets in improving the seismic behavior of structures, the effect of FRP sheets on the strength of structural members under lateral load (such as concrete columns), especially at high temperatures, has received less attention. Therefore, in this study, while making sure the accuracy of numerical simulations, then, by numerical modeling of three-dimensional finite element columns of reinforced concrete columns in both non-reinforced and reinforced conditions with CFRP sheets and thermal coupling-displacement analysis of this model during cyclic lateral loading at different fire temperatures, the effect of reinforcement of reinforced concrete columns with FRP sheets on the cyclic lateral behavior of these columns at different fire temperatures has been investigated. From the results of these studies, It has been observed that following the increase of fire temperature from 250 to 1000 ° C in both reinforced concrete columns and reinforced with FRP sheets, on the amount of lateral force, anchorage and energy of these columns during lateral loading of the cycle, the amount of lateral force, anchorage and energy of these columns during cyclic lateral bearing has been increased, which this increase (as the result of the increase in fire temperature) of the mentioned responses in the columns is more significant than the temperature of 500 ° C and especially in the reinforced concrete columns reinforced with FRP sheets.

In this work, a Semi-Algebraic Mode Analysis (SAMA) technique for multigrid waveform relaxation method applied to the finite element discretization on rectangular and regular triangular grids in two dimensions and cubic and triangular prism elements in three dimensions for the heat equation is proposed.  For all the studied cases especially for the general triangular prism element, both the stiffness and mass stencils are introduced comprehensively. Moreover, several numerical examples are included to illustrate the efficiency of the convergence estimates. Studying this analysis for the finite element method is more involved and more general than that finite-difference discretization since the mass matrix must be considered. The proposed analysis results are a very useful tool to study the behavior of the multigrid waveform relaxation method depending on the parameters of the problem.

In this paper, we solve a linear system of second-order boundary value problems by usingthe quadratic B-spline finite element method (FEM). The performance of the method is tested on one model problem. Comparisons are made with both the analytical solution and some recent results.The obtained numerical results show that the method is efficient.

This paper is concerned with the numerical treatment of delay reaction-diffusion with the Dirichlet boundary condition. The finite element method with linear B-spline basis functions is utilized to discretize the space variable. The Crank-Nicolson method is used for the processes of time discretization. Sufficient and necessary conditions for the numerical method to be asymptotically stable are investigated. The convergence of the numerical method is studied. Some numerical experiments are performed to verify the applicability of the numerical method.

In the present study, the Modified Equal Width (MEW) wave equation is going to be solved numerically by presenting a new technique based on the collocation finite element method in which trigonometric cubic B-splines are used as approximate functions. In order to support the present study, three test problems; namely, the motion of a single solitary wave, the interaction of two solitary waves, and the birth of solitons are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the numerical conserved laws as well as the error norms L2 and L∞.

In this paper, a combinatorial elliptic-circular waveguide is introduced to amplify electromagnetic waves. The cross-section of this waveguide is elliptic and filled by a dielectric material, whereas two axial circular hollows have been created in it. One of the hollows has been filled by an unmagnetized cold plasma and a relativistic pencil electron beam(RPEB) is injected inside other hollow. By applying an adaptive finite element method(FEM), electromagnetic slow waves amplification in the waveguide is investigated. We study variations of growth rate of excited microwaves under influence of different factors. The purpose of investigating the effect of various parameters of this waveguide such as plasma and electron beam radiuses, the RPEB location, dielectric constant and beam current intensity; is to introduce the waveguide with optimal configuration and parameters to obtain the highest wave growth rate.

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the twodimensional heat equation.

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B splines over finite elements. Propagation of a single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.

In this work, a semialgebraic mode analysis (SAMA) is proposed for investigating the convergence of a multigrid waveform relaxation method applied to the Finite Element (FE) discretization of the heat equation in two and three dimensions. This analysis for finite element methods is more involved and more general than that for Finite Difference (FD) discretizations, since mass matrix must be considered. The proposed analysis results in a very useful tool to study the behaviour of the multigrid waveform relaxation method depending on the parameters of the problem.

In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a space-time finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the so-called streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each space-time slab are solved by the generalized Newton-Raphson method. Exploiting the band-structured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each time-step a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.

In this paper, we are going to obtain the soliton solution of the generalized Rosenau-Kawahara-RLW equation that describes the dynamics of shallow water waves in oceans and rivers. We confirm that our new algorithm is energy-reserved and unconditionally stable. In order to determine the performance of our numerical algorithm, we have computed the error norms $L_{2}$ and $L_{infty }$. Convergence of full discrete scheme is firstly studied. Numerical experiments are implemented to validate the energy conservation and effectiveness for longtime simulation. The obtained numerical results have been compared with a study in the literature for similar parameters. This comparison clearly shows that our results are much better than the other results.

In this paper we consider the European continuous installment call option. Then its linear complementarity formulation is given. Writing the resulted problem in variational form, we prove the existence and uniqueness of its weak solution. Finally finite element method is applied to price the European continuous installment call option.