Optimal convergence rates for the combined methods of different finite element methods

Coupling techniques are essential to combining different numerical methods together for the purpose of solving an elliptic boundary value problem. By means of nonconforming constraints, the combinations of various Lagrange finite element methods often cause reduced rates of convergence. In this article, we present a method using penalty plus hybrid technique to match different finite element methods such that the optimal convergence rates in the ‖ · ‖_h and zero norms of errors of the solution can always be achieved. Also, such a coupling technique will lead to an optimal asymptotic condition number for the associated coefficient matrix. Moreover, this study can easily be extended for combining the finite difference method with the finite element method to also yield the optimal rate of convergence.     

Finite-element formulations of shear localization in granular materials

Displacement and mixed finite element formulations of shear localization in granular materials are presented. The formulations are based on hypoplastic constitutive laws for soils and the mixed-enhanced treatment involving displacement, strain and stress rates as independently varied fields. Included in these formulations are the standard displacement method, the three-field mixed formulation, the method of incompatible modes, the enhanced assumed strain method and the mixed enhanced strain method. Several numerical examples demonstrating the capability and performance of the different finite element formulations are presented. The numerical results are compared with available experimental data of Hostun RF sand and numerical results of Karlsruhe sand on biaxial tests. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori estimates for two multiscale finite element methods using multiple global fields to wave equations

We consider a scalar wave equation with nonseparable spatial scales. If the solution of the wave equation smoothly depends on some global fields, then we can utilize the global fields to construct multiscale finite element basis functions. We present two finite element approaches using the global fields: partition of unity method and mixed multiscale finite element method. We derive a priori error estimates for the two approaches and theoretically investigate the relation between the smoothness of the global fields and convergence rates of the approximations for the wave equation. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Numerical analysis of a PML model for time-dependent Maxwell’s equations

In this paper, we study a perfectly matched layer model for the three-dimensional time-dependent Maxwell’s equations. We develop both semi- and fully-discrete finite element methods for solving the truncated PML problem by Nedelec edge elements. Optimal convergence rates are proved for both semi- and fully-discrete schemes. To our knowledge, this is the first error analysis obtained for time domain finite element method for PML models.     

An Adaptive Local (AL) Basis for Elliptic Problems with Complicated Discontinuous Coefficients

We develop a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. In [5] a semidiscrete method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size H which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires O(log(1/H)^d+1) basis functions per mesh point where d denotes the spatial dimension of the computational domain. We present a fully discrete version of this method, where the AL basis is constructed by solving finite-dimensional localized problems, and which preserves the optimal convergence rates. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Summary. A least-squares mixed finite element method for general second-order non-selfadjoint elliptic problems in two- and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation and the flux approximation consist of piecewise polynomials of degree and respectively. The method is mildly nonconforming on the boundary. The cases and are studied. It is proved that the method is not subject to the LBB-condition. Optimal - and -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented. Received October 15, 1993 / Revised version received August 2, 1994     

PPIFE Method with Non-Homogeneous Flux Jump Conditions and Its Efficient Numerical Solver for Elliptic Optimal Control Problems with Interfaces

In this paper, we design a partially penalized immersed finite element method for solving elliptic interface problems with non-homogeneous flux jump conditions. The method presented here has the same global degrees of freedom as classic immersed finite element method. The non-homogeneous flux jump conditions can be handled accurately by additional immersed finite element functions. Four numerical examples are provided to demonstrate the optimal convergence rates of the method in $L^{\infty}$, $L^{2}$ and $H^{1}$ norms. Furthermore, the method is combined with post-processing technique to solve elliptic optimal control problems with interfaces. To solve the resulting large-scale system, block diagonal preconditioners are introduced. These preconditioners can lead to fast convergence of the Krylov subspace methods such as GMRES and are independent of the mesh size. Four numerical examples are presented to illustrate the efficiency of the numerical schemes and preconditioners.     

On the construction of optimal mixed finite element methods for the linear elasticity problem

Summary The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.     

Patch‐wise local projection stabilized finite element methods for convection–diffusion–reaction problems

In this article, we develop patch‐wise local projection‐stabilized conforming and nonconforming finite element methods for the convection–diffusion–reaction problems. It is a composition of the standard Galerkin finite element method, the patch‐wise local projection stabilization, and weakly imposed Dirichlet boundary conditions on the discrete solution. In this paper, a priori error analysis is established with respect to a patch‐wise local projection norm for the conforming and the nonconforming finite element methods. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates.     

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results from the discrete problem (3.1). Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations

Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the $p$ -version and the $h$ -version of the finite element method confirm the theoretical convergence rates.     

不可压缩流问题低次元稳定有限体积数值方法研究

分析了R^d,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈Hd,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈H¹情况下,本文方法得到的解与稳定有限元方法解之间具有O(h1情况下,本文方法得到的解与稳定有限元方法解之间具有O(h²)阶超收敛阶结果,且稳定有限体积方法取得了与稳定有限元方法相同的收敛速度,与稳定有限元方法比较,稳定有限体积方法计算简单高效,同时保持物理守恒,因此在实际应用中具有很好的潜力。     

Numerical analysis of finite element methods for miscible displacements in porous media

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element ( SUPG ) method is applied to the convection–diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 519–548, 1998     

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L ² and the H ¹ norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

SEMI-DISCRETE AND FULLY DISCRETE PARTIALPROJECTION FINITE ELEMENT METHODS FOR THEVIBRATING TIMOSHENKO BEAM

1.IntroductionTheTimoshenkobeammodelisgivenbywherethebeamisconsidereddamped,drepresentsthebeamthicknessandI~[0,1].000istherotationofverticalfibersinthebeamandw(x)istheverticaldisplacementofthebeam'scenterline(underaverticalloadgivenbyg(x)).Analogoust...     

关于一类二阶非线性双曲型方程全离散有限元方法的稳定性和收敛性估计

1　引　　言考虑如下混合问题 :φ( x,u) utt- di,j=1 xi( aij( x,u) u xj) - di=1bi( x,u) uxi =f( x,u)　　　　　　　　　　　　　　　 ( x,t)∈Ω× [0 ,T]u( x,0 ) =0 ,　 ut( x,0 ) =0　　 x∈Ωu( x,t) =0　 ( x,t)∈ Ω× [0 ,T]( 1 .1 )这里 utt= 2 u t2 ,uxi= u xi;Ω是 Rd 中充分光滑的有界开域 ,边界 Ω光滑 .对于 φ( x,u)中仅含有 x,或 φ( x,u)≡ 1的线性或非线性双曲型方程的半离散或全离散有限元方法的研究已有 [1 ] -[4 ] .这些文献定义了线性[1 ] [4] 或非线性[2 ] 或预测—校正[4] 全离散有限元格式 ,然后将原方…     

Two temporal second-order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations

In this paper, a special point is found for the interpolation approximation of the distributed order fractional derivatives to achieve at least second-order accuracy. Then, two H¹-Galerkin mixed finite element schemes combined with the higher accurate interpolation approximation are introduced and analyzed to solve the distributed order fractional sub-diffusion equations. The stable results, which just depend on initial value and source item, are derived. Some a priori estimates with optimal order of convergence both for the unknown function and its flux are established rigorously. It is shown that the H¹-Galerkin mixed finite element approximations have the same rates of the convergence as in the classical mixed finite element method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, some numerical experiments are presented to show the efficiency and accuracy of H¹-Galerkin mixed finite element schemes.     

A mixed finite element method for solving the nonstationary stokes equation

Summary This paper deals with a mixed finite element method for approximating a fourth order initial value problem arising from the nonstationary Stokes problem. For piecewise linear shape functions error estimates are given with convergence rates similar to the elliptic case. Some numerical computations will illustrate the theoretical results.