Two-level Finite Element Method Research Articles

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

Two-level finite element methods for the steady bio-convection flows problem

Two-level finite element methods for the steady bio-convection flows problem

Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. The AB method is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size. We provide analysis to show how the AB method mitigates the pollution error.

Stabilizing the convection–diffusion–reaction equation via local problems

This paper presents a novel two-level finite element method for convection–diffusion–reaction problems. The proposed scheme consists of a global problem and an ensemble of local problems. The boundary conditions of the local problems are provided by a global approximation of the same problem. The stability is ensured with an artificial diffusion mechanism that acts on the difference between local and global approximation, and leads to an a priori error estimate for the global solution. For the computation of the solutions, an efficient fixed-point algorithm is proposed as an alternative to the well established static condensation technique. The local solutions can be computed in parallel and enter in a communication step. This step takes place in the entire domain and consists of a mass matrix inversion only. Therefore, the overall algorithm is easy to implement and computationally inexpensive.

Two-level Picard coupling correction finite element method based on charge-conservation for stationary inductionless magnetohydrodynamic equations

In this paper, we propose two two-level finite element methods based on charge-conservation to solve the 2D/3D stationary inductionless magnetohydrodynamic (MHD) equations in Lipschitz domain. The methods are based on the Picard coupling correction on the fine mesh, one has to solve a nonlinear problem on the coarse mesh, the other one need to solve a linearized problem with Newton iteration on the coarse mesh. In addition, we give the stabilities and error estimates for the proposed methods. Several numerical results are presented to verify the theoretical analysis and show the effectiveness of the methods.

A Two-Level Nonconforming Rotated Quadrilateral Finite Element Method for the Stationary Navier–Stokes Equations

In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective.

Error Analysis of Two-Level Finite Element Method for the Nonlinear Conductivity Problem in Maxwell's System

Error Analysis of Two-Level Finite Element Method for the Nonlinear Conductivity Problem in Maxwell's System

Composite coating effect on stress intensity factors of aluminum pressure vessels with inner circumferential crack by X-FEM

In this paper, the fracture behavior of a pressure vessel involving a full circumferential wall crack at the inner surface is investigated by applying the axisymmetric model. By using the extended finite element method (X-FEM), the stress intensity factor is calculated and compared for the case of a crack in the Aluminum monolayer cylinder as well as when the composite coating is mounted on its outer surface. Also, the stress intensity factor is obtained for different crack lengths, materials, and fiber orientations in the composite laminate. The materials of composite laminate are Glass-epoxy and Carbon-epoxy. An axisymmetric extended finite element formulation is developed for the pressure vessel, and a MATLAB programming code is implemented to solve the problem. To validate the developed code, two examples are solved and compared with the available results of two-level finite element method (2LFEM) and exact solutions. The results obtained from the axisymmetric X-FEM model is in good agreement with the existing results. The results show that the most reduction of SIFs occurred in pressure vessels with the Carbon-epoxy coating with 0° fiber orientation.

A two-level finite element method with backtracking technique for the Navier-Stokes equations at high Reynolds numbers

A two-level finite element method with backtracking technique for the stationaryNavier-Stokes equations at high Reynolds numbers is considered. The basic idea of the method is to first solve a fully nonlinearNavier-Stokes equations with a subgrid stabilization term on a coarse grid, and then solve a subgrid stabilized linear fine grid problem based on one step of Newton iteration,and finally, solve a linear correction problem on the coarse mesh. The theoretical and numerical results show that, with suitable scalings of algorithmic parameters,the method can yield an optimal convergence rate. Numerical results are also given to verify theoretical predictions and demonstrate the effectiveness of the method.

Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively.

A two-level finite element method for the Allen–Cahn equation

ABSTRACTWe consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.

Iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem

In the paper we consider the one-level and two-level iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem based on the iteration of pressure with a factor of penalty parameter. Firstly, the $$\mathbf {H}^1$$ and $$\mathbf {L}^2$$ error estimates of numerical solutions of one-level iterative penalty finite element method are provided. Secondly, the stability and convergence of two-level iterative penalty finite element method are analyzed. Finally, some numerical results are provided to verify the effectiveness of the developed numerical schemes.

Error estimates of two-level finite element method for Smagorinsky model

Two-level finite element method for simulating Smagorinsky model in large eddy simulation is investigated. In this two-level algorithm, a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh. Corresponding to Newton linearization method, the linearized Smagorinsky model is solved on the fine mesh. The error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in Smagorinsky model, the two-level method provides the optimal convergence rates for the velocity in H1 norm and the pressure in L2 norm. Meanwhile, the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two-level algorithm.

Semi-coarsening AMLI preconditioning of anisotropic trilinear FEM systems

Optimal order algebraic multilevel iteration (AMLI) preconditioners based on recursive application of two-level finite element (FE) methods and polynomial stabilization have been introduced and analyzed in Axelsson and Vassilevski (1989, 1990). The construction follows the natural hierarchical splitting using the fact that the finite element spaces corresponding to two successive mesh refinements are nested. Uniform estimates for the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality are very important for the derivation of optimal order methods. The value of the upper bound for γ∈(0,1) is a part of the construction of the multilevel extension of the related two-level method.In this paper algebraic two-level and multilevel preconditioning algorithms for second order anisotropic elliptic boundary value problems are constructed. Here we allow big jumps in the coefficients and varying the direction of dominating anisotropy from one element to another in the coarse triangulation. The discretization is done by trilinear conforming finite elements where the semi-coarsening mesh refinement strategy is applied. A new uniform estimates for the related CBS constants are derived. The additive preconditioning strategy for the system with the pivot block in the hierarchical two-level splitting is proposed, where the related sub-problems have two dimensional structure.

Determination of stress intensity factors by the finite element discretized symplectic method

When rewriting the governing equations in Hamiltonian form, analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions. In theory, there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions. In practice, the matrix relating the coefficients to the outer boundary conditions is ill-conditioned unless the boundary is very simple, e.g., circular. In this paper, a new two-level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed. With the available classical finite elements and symplectic series, the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series. Since the first few coefficients are the stress intensity factors, post-processing is not required. A number of numerical examples as well as convergence studies are given.

Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations

The two-level penalty mixed finite element method for the stationary Navier-Stokes equations based on Taylor-Hood element is considered in this paper. Two algorithms are proposed and analyzed. Moreover, the optimal stability analysis and error estimate for these two algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms.

Two-Level Iteration Penalty Methods for the Navier-Stokes Equations with Friction Boundary Conditions

This paper presents two-level iteration penalty finite element methods to approximate the solution of the Navier-Stokes equations with friction boundary conditions. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh sizeHin combining with solving a Stokes, Oseen, or linearized Navier-Stokes type variational inequality problem for Stokes, Oseen, or Newton iteration on a fine mesh with mesh sizeh. The error estimate obtained in this paper shows that ifH,h, andεcan be chosen appropriately, then these two-level iteration penalty methods are of the same convergence orders as the usual one-level iteration penalty method.

Error estimate for a simple two-level discretization of stream function form of the Navier–Stokes equations

In this paper, a simplified two-level finite element method with backtracking is proposed for the stream function formulation of the Navier–Stokes equations. This method requires the solution of two small systems on the coarse space and one symmetric, positive linear problem on the refined space. Error analysis for the case of Clough–Tocher or Bogner–Fox–Schmit elements is presented and the optimal asymptotic estimation is obtained in the H2- and H1-norms. Finally, numerical examples are given to verify the theoretical analysis.

Analysis of Mode III Elastodynamic Cracked Plane Using the Fractal Two-Level Finite Element Method

In this article, the fractal two-level finite element method, which has mainly been used for static crack problems, is applied to mode III elastodynamic plane crack problems. Using the transformation process in the proposed method, the infinite dimension of the finite element matrices that are assembled for a singular region is made finite in terms of the dynamics stress intensity factors directly and, thus, the computational time and errors can be reduced significantly. The Newmark time integration scheme is then used to obtain the dynamic stress intensity factors for different crack cases. The results from the proposed method are in reasonable agreement with those of classical methods. A parametric study is conducted to investigate the effects of different parameters on the accuracy of the dynamic stress intensity factor.

Two-Level Finite Element Method with a Stabilizng Subgrid for the Natural Convention Flow Simulation in Different Geometries

The present numerical study deals with the stabilized finite element solution of the steady natural convection flow in terms of primitive variables. The domain is discretized into a set of regular linear triangular elements for all unknowns containing a single subgrid node, which has a crucial effect in the stabilization with a cheap computational cost. The proposed stabilized finite element method enables obtainment of a stable solution and avoids oscillations, especially in the pressure. The algorithm is presented for different benchmark problems considered in different geometries and the solutions are obtained for high values of Rayleigh number (Ra) (103 ≤ Ra ≤ 106). Furthermore, the solutions show the well-known characteristics of the natural convection flow and agree well with the results in the literature.