Semidiscrete Galerkin method for equations of motion arising in Kelvin‐Voigt model of viscoelastic fluid flow

Finite element Galerkin method is applied to equations of motion arising in the Kelvin–Voigt model of viscoelastic fluids for spatial discretization. Some new a priori bounds which reflect the exponential decay property are obtained for the exact solution. For optimal L^∞( L ²) estimate in the velocity, a new auxiliary operator which is based on a modification of the Stokes operator is introduced and analyzed. Finally, optimal error bounds for the velocity in L^∞( L ²) as well as in L^∞( H )‐norms and the pressure in L^∞(L²)‐norm are derived which again preserves the exponential decay property. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.     

Superconvergence of finite element approximations to Maxwell's equations

We study superconvergence of edge finite element approximations to the magnetostatic problem and to the time-dependent Maxwell system. We show that in special discrete norms there is an increase of one power in the order of convergence of the finite element method compared to error estimates in standard Sobolev norms. Our results are restricted to an orthogonal grid in R ³, but the grid may be nonuniform. © 1994 John Wiley & Sons, Inc.     

Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Stabilized discontinuous finite element approximations for Stokes equations

In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite element spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L²$L^2$ norm are obtained, as well as one in a mesh-dependent norm for the velocity in both symmetric and nonsymmetric formulations.     

Convergence of finite element approximations of large eddy motion

Fluid motion in many applications occurs at higher Reynolds numbers. In these applications dealing with turbulent flow is thus inescapable. One promising approach to the simulation of the motion of the large structures in turbulent flow is large eddy simulation in which equations describing the motion of local spatial averages of the fluid velocity are solved numerically. This report considers “numerical errors” in LES. Specifically, for one family of space filtered flow models, we show convergence of the finite element approximation of the model and give an estimate of the error. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 689–710, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10027.     

A weak Galerkin finite element method for the Oseen equations

In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i ? 1 for the pressure and enhancing the polynomials of degree i ? 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem.     

A weak Galerkin finite element method for the stokes equations

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.     

Sobolev方程的时间间断Galerkin有限元方法

引入Sobolev方程的等价积分方程,构造Sobolev方程的新的时间间断Galerkin有限元格式.该格式不仅保持有限元解在时间剖分点处的间断特性,而且避免了传统时空有限元格式中跳跃项的出现,从而降低了格式理论分析和数值模拟的复杂性.证明了Sobolev方程的时间间断而空间连续的时空有限元解的稳定性、存在唯一性、L2...     

Error analysis of finite element approximations of the stochastic Stokes equations

Numerical solutions of the stochastic Stokes equations driven by white noise perturbed forcing terms using finite element methods are considered. The discretization of the white noise and finite element approximation algorithms are studied. The rate of convergence of the finite element approximations is proved to be almost first order (h|ln h|) in two dimensions and one half order ( h^\frac12h^{\frac{1}{2}}) in three dimensions. Numerical results using the algorithms developed are also presented.     

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β ∈ (0, 1) and 2α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax-Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.     

Some upwinding techniques for finite element approximations of convection-diffusion equations

Summary A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as artificial diffusion terms. These artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices. Three basis matrices are symmetric, rank one diffusion operators associated with the edges of the triangle; the fourth basis matrix is skew symmetric and is associated with a rotation by /2. While finite volume discretizations may be written as upwinded Galerkin methods, the converse does not appear to be true. Our approach is used to examine several upwinding schemes, including the streamline diffusion method, the box method, the Scharfetter-Gummel discretization, and a divergence-free scheme.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440The work of this author was supported through KWF-Landis/Gyr Grant 1496, AT & T Bell Laboratories, and Cray Research     

A hybridized weak Galerkin finite element scheme for the Stokes equations

In this paper a hybridized weak Galerkin(HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced.The WG method uses weak functions and their weak derivatives which are defined as distributions.Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees.Different combination of polynomial spaces leads to different WG finite element methods,which makes WG methods highly flexible and efficient in practical computation.A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution.With this new feature,the HWG method can be used to deal with jumps of the functions and their flux easily.Optimal order error estimates are established for the corresponding HWG finite element approximations for both primal variables and the Lagrange multiplier.A Schur complement formulation of the HWG method is derived for implementation purpose.The validity of the theoretical results is demonstrated in numerical tests.     

Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains

Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.     

Error estimates for a stabilized finite element method for the Oldroyd B model

In this paper, we study a new approximation scheme of transient viscoelastic fluid flow obeying an Oldroyd-B-type constitutive equation. The new stabilized formulation bases on the choice of a modified Euler method connected to the streamline upwinding Petrov-Galerkin (SUPG) method [M. Bensaada, D. Esselaoui, D. Sandri, Stabilization method for continuous approximation of transient convection problem, Numer. Methods Partial Differential Equations 21 (2004) 170-189], in order to stabilize the tensorial transport term of the Oldroyd derivative. Suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. A priori error estimates for the approximation in terms of the mesh parameter h and the time discretization parameter Δt are derived.     

Two-grid methods for finite volume element approximations of nonlinear parabolic equations

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H³|lnH|)$h=O\left(H^3|lnH|\right)$. As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.