Convergence Of Finite Element Method Research Articles

Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection–diffusion problem with exponential layers

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order k under an energy norm for finite element method of any order k. Numerical experiments validate our theoretical findings.

A fast linearized Galerkin finite element method for the nonlinear multi-term time fractional wave equation

This paper is concerned with a class of nonlinear multi-term time fractional wave equations. Based on the Galerkin finite element method (FEM) in space and the L2-1σ formula in time, combined with the corresponding fast algorithm, an efficient fully discrete scheme is constructed. The unconditional optimal convergence property in H1-norm of the proposed scheme is discussed by utilizing the time-space splitting technique. Furthermore, the optimal L2-norm convergence and the H1-norm superconvergence results of the method are obtained. Finally, several illustrative numerical experiments are performed to verify the theoretical results. Compared with the existing works, studying the convergence of FEMs for the nonlinear fractional wave equation is a great challenging task, and there are significant differences in research approaches.

Optimal Strong Convergence of Finite Element Methods for One-Dimensional Stochastic Elliptic Equations with Fractional Noise

Optimal Strong Convergence of Finite Element Methods for One-Dimensional Stochastic Elliptic Equations with Fractional Noise

Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.

Uniform convergence of finite element methods on Bakhvalov-type meshes in the case of N−1 ≤ ε

It has been reported that convergence stalls on Bakhvalov-Shishkin mesh in the case of N−1≤ε, where ε is the singular perturbation parameter and N is the number of mesh intervals. To analyze these phenomena, we present uniform convergence analysis of finite element methods on Bakhvalov-type meshes, one popular kind of graded meshes closely related to Bakhvalov-Shishkin mesh, when N−1≤ε. An optimal order of convergence is proved and this result is used for the improvement of Bakhvalov-Shishkin mesh. These theoretical results are verified by numerical experiments.

The optimal convergence of finite element methods for the three-dimensional second-order elliptic boundary value problem with singularity

Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 for approximations with a Pk finite element method.

Convergence of finite element methods for hyperbolic heat conduction model with an interface

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures.

Single variable shear deformation theory for free vibration and harmonic response of frames on flexible foundation

A single variable shear deformation theory (SVSDT) is applied to free vibrations and harmonic response analysis of a multi-storey frame model considering flexibility of foundation. The column-foundation joints are modelled using three-spring approach instead of fixed support assumption to show the effects of soil-structure interaction (SSI) on the dynamic behaviour of frame model. The dynamic stiffness formulations are used for both free vibration and harmonic response analysis. The results that obtained using SVSDT and Timoshenko beam theory (TBT) are presented comparatively. The calculated natural frequencies are verified by finite element method (FEM). The convergence of FEM for the first three natural frequencies is presented for the fixed supported frame and frame considering SSI. The harmonic response curves are used to reveal the resonant and anti-resonant frequencies directly. It is seen that using a lateral displacement response for harmonic response curves may cause to miss natural frequencies of axial dominant modes of the frame model. However, all required natural frequency values of multi-storey frames considering foundation flexibility can be obtained via harmonic response curves that plotted by using a vertical displacement response of a node of model. Moreover, the proposed approach is validated by experimental data for scaled frame models, which are fixed supported and flexibly supported, respectively. The computation times for harmonic responses are tabulated for frame model on flexible foundation as well as fixed support assumption.

Grinding temperature field prediction by meshless finite block method with double infinite element

Simulation is an important method to investigate grinding temperature and prevent unwanted heat damages of workpiece by excessive grinding heat. Most previous numerical studies on grinding process analysis were based on finite element method, however, meshing is an arduous task especially for a complex geometry, and the convergence of finite element method has been proved to be bad in some cases. To address this, more numerical algorithms with higher adaptiveness and lower cost have been proposed such as meshless formulation. Meshless finite block method with double infinite element is a numerical method developed from meshless method. This method has higher accuracy and convergence. The grinding model is closer to the real fact. The mapping technique is used to transform a block of quadratic type from Cartesian coordinate (x, y) to normalized coordinate (ξ, η) with three or five seeds for double or single infinite element. The Lagrange series approximation is applied to construct differential matrices in normalized domain with nodes following Chebyshev's roots. The static and transient heat transfer processes were simulated by meshless finite block method with double infinite element, and a better convergence was demonstrated by comparison with the finite element method (ABAQUS). This study has also proved that the convergence depended on the workpiece feed velocity for both the present method and the finite element method. In addition, six different types of heat source were applied on simulating the grinding temperature field of titanium alloy TC4 and compared with experimental results, which shows that simulated result with triangular distribution heat source showed a good agreement with that of experiments.

Unconditional error estimates for time dependent viscoelastic fluid flow

The unconditional convergence of finite element method for two-dimensional time-dependent viscoelastic flow with an Oldroyd B constitutive equation is given in this paper, while all previous works require certain time-step restrictions. The approximation is stabilized by using the Discontinuous Galerkin (DG) approximation for the constitutive equation. The analysis bases on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element approximation of corresponding iterated time-discrete PDEs. The approach used in this paper can be applied to more general couple nonlinear parabolic and hyperbolic systems.

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

SummaryLinear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.

Well-posedness and a numerical study of a regularization model with adaptive nonlinear filtering for incompressible fluid flow

We give a detailed analytical study of a Leray model of incompressible flow that uses nonlinear filtering based on indicator functions. The indicator functions allow for local regularization, instead of global regularization which can over-smooth and dampen out important flow structures. The key to the analysis is the identification of the indicator function as a Nemyskii operator. After proving well-posedness, we provide a numerical study which includes proving optimal convergence of finite element method for the model, as well as several numerical experiments.

Dynamic Analysis of AP1000 Shield Building Considering Fluid and Structure Interaction Effects

The shield building of AP1000 was designed to protect the steel containment vessel of the nuclear reactor. Therefore, the safety and integrity must be ensured during the plant life in any conditions such as an earthquake. The aim of this paper is to study the effect of water in the water tank on the response of the AP1000 shield building when subjected to three-dimensional seismic ground acceleration. The smoothed particle hydrodynamics method (SPH) and finite element method (FEM) coupling method is used to numerically simulate the fluid and structure interaction (FSI) between water in the water tank and the AP1000 shield building. Then the grid convergence of FEM and SPH for the AP1000 shield building is analyzed. Next the modal analysis of the AP1000 shield building with various water levels (WLs) in the water tank is taken. Meanwhile, the pressure due to sloshing and oscillation of the water in the gravity drain water tank is studied. The influences of the height of water in the water tank on the time history of acceleration of the AP1000 shield building are discussed, as well as the distributions of amplification, acceleration, displacement, and stresses of the AP1000 shield building. Research on the relationship between the WLs in the water tank and the response spectrums of the structure are also taken. The results show that the high WL in the water tank can limit the vibration of the AP1000 shield building and can more efficiently dissipate the kinetic energy of the AP1000 shield building by fluid-structure interaction.

The Analysis of Convergence of Finite Element Method for a Foundation Slab on the Bilinear Subgrade

This article considers the convergence of the foundation slab finite element model. Slab works in conjunction with the subgrade. In PC Cross Scad authors conducted joint calculation of the foundation slab and ground, based on the use of an iterative algorithm.Analytical research of convergence of the calculations results allowed to select optimum partitioning step of the model. Conclusions are drawn on rationality of calculations of the foundation slab on the bilinear subgrade realized in PC Cross Scad.

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h 2 + τ k ) in the energy norm and O((h 2 + τ k )(h 2 + τ)) in the L 2(Q)-norm.

Finite element method for a class of parabolic integro-differential equations with interfaces

In this paper, convergence of finite element method for a class of parabolic integro-differential equations with discontinuous coefficients are analyzed. Optimal L2(L2) and L2 (H1) norms are shown to hold when the finite element space consists of piecewise linear functions on a mesh that do not require to fit exactly to the interface. Both continuous time and discrete time Galerkin methods are discussed for arbitrary shape but smooth interfaces.

Weak Convergence of Finite Element Method for Stochastic Elastic Equation Driven By Additive Noise

In this paper we study the weak convergence of the semidiscrete and full discrete finite element methods for the stochastic elastic equation driven by additive noise, based on \(C^0\) or \(C^1\) piecewise polynomials. In order to simplify the analysis of weak convergence, we rewrite the stochastic elastic equation in an abstract problem and the solutions of the semidiscrete and full discrete problems in a unified form. We obtain that the weak order is twice the strong order, both in time and in space. Numerical experiments are carried out to verify the theoretical results.

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

AbstractA finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L∞(L2) and L∞(H1) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L∞(H1) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Finite element approximation for time-dependent diffusion with measure-valued source

The convergence of finite element methods for elliptic and parabolic partial differential equations is well-established if source terms are sufficiently smooth. Noting that finite element computation is easily implemented even when the source terms are measure-valued—for instance, modeling point sources by Dirac delta distributions—we prove new convergence order results in two and three dimensions both for elliptic and for parabolic equations with measures as source terms. These analytical results are confirmed by numerical tests using COMSOL Multiphysics.

Rehabilitation of the Lowest-Order Raviart–Thomas Element on Quadrilateral Grids

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429–2451] reveals that convergence of finite element methods using $H(\mathrm{div}\,,\Omega)$-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart–Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and least-squares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart–Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation.