Finite Element Subspaces Research Articles

Computation of Single Eigenfrequencies and Eigenfunctions of Plate and Shell Structures Using an H-adaptive FE-method

The accuracy of the computation of eigenfrequencies and eigenfunctions with FE-methods can be substantially improved with efficient adaptive procedures. For such an adaptive analysis of plate and shell structures a simple a-posteriori error estimator or indicator for the error in the energy norm and L2-norm of the eigenfunction uh for shell and plate structures is proposed.Both indicators hite shall represent the correct convergence of the estimated error. The estimator for the error in the energy norm is used to enlarge adaptively the dimension of the finite element subspace. On the basis of numerical examples the efficiency and the quality of the improved solution is discussed. In order to validate the quantity of the estimated error Aitken’s extrapolation technique is applied.

A NEW APPROXIMATE INERTIAL MANIFOLD AND ASSOCIATED ALGORITHM

In this article the authors propose a new approximate inertial manifold(AIM) to the Navier-Stokes equations. The solutions are in the neighborhoods of this AIM with thickness δ = o( h 2k+1–E). The article aims to investigate a two grids finite element approximation based on it and give error estimates of the approximate solution ∥ | ( u - u h * *, p - p h * * ) ∥ | ≤ C ( h 2 k + 1 - ɛ + h * ( m + 1 ) where ( h, h*) and ( k, m) are coarse and fine meshes and degree of finite element subspaces, respectively. These results are much better than Standard Galerkin(SG) and nonlinear Galerkin (NG) methods. For example, for 2D NS eqs and linear element, let u h, u h, u* be the SG, NG and their approximate solutions respectively, then ∥ u - u h ∥ 1 ≤ C h, ∥ u - u h ∥ 1 ≤ C h 2, ∥ u - u * ∥ 1 ≤ C h 3, and h* ≈ h 2 for NG, h* ≈ h 3/2 for theirs.

Optimality of Multilevel Preconditioners for Local Mesh Refinement in Three Dimensions

In this article, we establish optimality of the Bramble--Pasciak--Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D red-green result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D red-green refinement procedure introduced by Bornemann, Erdmann, and Kornhuber, and then to use this result to extend the WHB optimality results from the quasi-uniform setting to local 2D and 3D red-green refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction turns out to rest not only on the shape regularity of the refined elements, but also critically on a number of geometrical properties we establish between neighboring simplices produced by the Bornemann--Erdmann--Kornhuber (BEK) refinement procedure. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical, implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H1-stability of the L2-projection. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied. The theoretical framework employed supports arbitrary spatial dimension $d \geq 1$ and requires no coefficient smoothness assumptions beyond those required for well-posedness in H1.

An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems

This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones and Vassilevski 2001. Then, the algebraic construction from Jones, Vassilevski and Woodward 2003 of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. A numerical illustration is also provided.

Enriched finite element subspaces for dual–dual mixed formulations in fluid mechanics and elasticity

In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dual–dual ones), which are analyzed by means of a slight generalization of the well known Babuška–Brezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, Raviart–Thomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces.

A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part II: a posteriori error analysis

This is the second part of a work dealing with a low-order mixed finite element method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. In the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation, and that the corresponding Galerkin scheme becomes well posed with piecewise constant functions and Raviart–Thomas spaces of lowest order as the associated finite element subspaces. In this paper we develop a Bank–Weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions. Several numerical results illustrating the efficiency of the method are also provided.

Least-Squares Methods for Linear Elasticity

This paper develops least-squares methods for the solution of linear elastic problems in both two and three dimensions. Our main approach is defined by simply applying the L2 norm least-squares principle to a stress-displacement system: the constitutive and the equilibrium equations. It is shown that the homogeneous least-squares functional is elliptic and continuous in the $H({\rm div};\,\Omega)^d \times H^1(\Omega)^d$ norm. This immediately implies optimal error estimates for finite element subspaces of $H({\rm div};\,\Omega)^d \times H^1(\Omega)^d$. It admits optimal multigrid solution methods as well if Raviart--Thomas finite element spaces are used to approximate the stress tensor. Our method does not degrade when the material properties approach the incompressible limit. Least-squares methods that impose boundary conditions weakly and use an inverse norm are also considered. Numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our ...

High‐order vector elements for finite element electromagnetic analysis and subspace method in large‐scale eigenvalue problems

AbstractIn three‐dimensional electromagnetic field analysis by the finite element method, the function space of the higher‐order vector elements is separated into the gradient field and the rotational field and a new configuration method for the shape function is proposed. For a large‐dimensional eigenvalue problem with unique zero eigenvalue in the electromagnetic eigenvalue problem, the subspace method containing an algorithm eliminating the null space is proposed. When these methods are applied to the analysis of a cubic cavity resonator, fast analysis of a large‐dimensional eigenvalue problem becomes possible. In addition, it is found that more accurate solutions can be obtained with the higher‐order vector elements in the same computation time. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 85(7): 57–65, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.10017

An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

We consider the coupling of dual-mixed finite element and boundary element methods to solve a linear-nonlinear transmission problem in plane hyperelasticity with mixed boundary conditions. Besides the displacement and the stress tensor, we introduce the strain tensor as an additional unknown, which yields a two-fold saddle point operator equation as the corresponding variational formulation. We derive a reliable a posteriori error estimate that depends on the solution of local Dirichlet problems and on residual terms on the transmission and Neumann boundaries, which are given in a negative order Sobolev norm. Our approach does not need the exact Galerkin solution, but any reasonable approximation of it. In addition, the analysis does not depend on special finite element or boundary element subspaces. However, for certain specific subspaces we are able to provide two fully local a posteriori error estimates, in which the residual terms are bounded by weighted local L2-norms. Further, one of the error estimates does not require the explicit solution of the local problems.

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux σ , we introduce ∇ u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.

Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) and H ( Div ; Ω ) {{\boldsymbol H}(\operatorname {Div};\Omega )} . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain Ω \Omega , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for H ( Div ; Ω ) \boldsymbol H(\operatorname {Div};\Omega ) and H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: div \operatorname {div} , c u r l \operatorname {\mathbf {curl}} and g r a d \operatorname {\mathbf {grad}} can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar H ( Div ; Ω ) \boldsymbol H (\operatorname {Div};\Omega ) - and H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity-stress-pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H1 product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L2-norm and in the H1-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

First-Order System Least Squares (FOSLS) for Convection-Diffusion Problems: Numerical Results

The focus of this paper is on planar linear convection-diffusion problems, to which we apply a special form of first-order system least squares (FOSLS [Cai et al., SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799; Cai, Manteuffel, and McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425--454]). This we do by introducing the gradient of the primary variable, scaled by certain exponential functions. The convection-diffusion equation is then recast as a minimization principle for a functional corresponding to a sum of weighted L2 norms of the resulting first-order system. Discretization is accomplished by a Rayleigh--Ritz method based on standard finite element subspaces, and the resulting linear systems are solved by basic multigrid algorithms. The main goal here is to obtain optimal discretization accuracy and solver speed that is essentially uniform in the size of convection. Our results show that the FOSLS approach achieves this goal in general when the performance is measured either by the functional or by an equivalent weighted H1 norm. Included in our study is a multilevel adaptive refinement method based on locally uniform composite grids and local error estimates based on the functional itself.

A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems

We consider a nonconforming streamline-diffusion finite element method for solving convection-diffusion problems. The theoretical and numerical investigation for triangular and tetrahedral meshes recently given by John, Maubach and Tobiska has shown that the usual application of the SDFEM gives not a sufficient stabilization. Additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case. The error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space. Thus, the analysis can be applied for example to the Crouzeix/Raviart element but not to the nonconforming quadrilateral elements proposed by Rannacher and Turek. In this paper, parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case. Numerical experiments support the theoretical predictions.

Least-squares for second-order elliptic problems

In this paper, we introduce and analyze two least-squares methods for second-order elliptic differential equations with mixed boundary conditions. These methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution. With the methods to be developed, Neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite element subspaces. The resulting least-squares approximations are unconditionally stable (no conditions on the step-size h) and will be shown to converge at an optimal rate. The first least-squares method involves a discrete, computable H −1-norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise L 2-norm of the residual over the finite elements. This method is developed without the introduction of additional problem variables. The second method involves the use of the flux as an additional unknown. Although this method is similar to the least-squares method for first-order systems introduced in [7], it differs in that discontinuous finite elements are allowed. It is also more general in that it extends to the oblique boundary problem.

Boundary-hybrid finite elements and a posteriori error estimation

Finite element methods that calculate the normal derivative of the solution along the mesh interfaces and recover the solution via local Neumann problems were introduced about two decades ago by I, Babus̆ka, J.T. Oden and J.K. Lee for the treatment of the homogeneous Laplace equation and called ‘boundary-hybrid method’. We revisit this approach for general symmetric and positive definite elliptic equations with homogeneous boundary conditions. The resulting approximation is nonconforming, and the corresponding error is orthogonal to all the conforming finite element subspaces. This crucial property shows immediately how to derive an a posteriori error estimator for conforming finite element approximations via Pythagoras' theorem. The investigation of this idea leads to a sound strategy for a posteriori error analysis which gives conservative, yet accurate, estimates, is cheap for good conforming approximations, and otherwise produces an enhanced solution at not significantly more than the cost normally expected for such a result.

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.

Strong-norm error estimates for the projective-difference method for approximately solving abstract parabolic equations

Solutions continuously differentiable with respect to time of parabolic equations in Hilbert space are obtained by the projective-difference method approximately. The discretization of the problem is carried out in the spatial variables using Galerkin's method, and in the time variable using Euler's implicit method. Strong-norm error estimates for approximate solutions are obtained. These estimates not only allow one to establish the convergence of the approximate solutions to the exact ones but also yield numerical characteristics of the rates of convergence. In particular, order-sharp error estimates for finite element subspaces are obtained.

Reduction of finite element subspace dimension based on estimated error

This paper examines the idea of basing dimensional reduction for finite element subspaces on estimated error. Orthonormal bases related to the error terms in a quadratic error estimator are established by solving algebraic eigenvalue problems, to identify components most suitable for deletion from finite element subspaces. Error-based eigenvectors are compared with less expensive vibration eigenvectors for two example problems. It is found that vibration eigenvectors correlate very well with error-based eigenvectors, so that optimal dimensional reduction can be approximated very well using traditional modal truncation. In addition, results indicate that substantial participation of higher vibration eigenvectors in a finite element approximation can be indicative of a high level of error.

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in $n=2$ or 3 dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the $H(\divv) \times H^1$ norm and to yield optimal convergence for finite element subspaces of $H(\divv) \times H^1$. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the $(H^1)^{n+1}$ norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of $(H^1)^{n+1}$. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection--diffusion--reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.