A Note on the Energy Norm for a Singularly Perturbed Model Problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research. Received October 25, 2001; revised July 12, 2002 Published online: October 24, 2002     

Suboptimal anisotropic filtering in a finite horizon

Consideration was given to the problem of robust stochastic filtering in a finite horizon for the linear discrete time-varying system. A random disturbance with inaccurately known probabilistic distribution is fed to the system input. Uncertainty of the input disturbance is defined in the information-theoretical terms by the anisotropy functional of a random vector. The sufficient condition for strict boundedness of the anisotropic norm of linear discrete timevarying system assigned by the threshold value (lemma of real boundedness) was proved in terms of the matrix inequalities. Sufficient conditions for boundedness of the anisotropic norm of two limiting cases of the anisotropy levels of the input disturbance (a = 0 and a → ∞) were established. A sufficient existence condition for the estimator guaranteeing boundedness of the anisotropic norm of the estimation error operator by the given threshold value was formulated and proved. Sufficient existence conditions for the estimators of two limiting cases of the anisotropy levels of input disturbance were obtained. The estimation algorithm relies on the recurrent solution of a system of matrix inequalities.     

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.     

Adaptivity and a Posteriori Error Control for Bifurcation Problems?II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier?CStokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z ₂ symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual?CWeighted?CResidual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.     

An adaptive mesh refinement of quadrilateral finite element meshes based upon a posteriori error estimation of quantities of interest: linear static response

A posteriori error estimation in finite element analysis serves as an important guide to the meshing tool in an adaptive refinement process. However, the traditional posteriori error estimates, which are often defined in the energy or energy-type norms over the entire domain, provide users insufficient information regarding the accuracy of specific quantities in the solution. This paper describes an adaptive quadrilateral refinement process with a goal-oriented error estimation, in which a posteriori error is estimated with respect to the specified quantity of interest. A highlight of this paper is the demonstration of tools described in the paper used in a practical industrial environment. The performance of this process is demonstrated on several practical problems where the comparison is with the adaptive process based on the traditional error estimation.     

Adaptive Finite Element Methods for the Identification of Elastic Constants

In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.     

A posteriori error estimates for the primary and dual variables for the div first-order least-squares finite element method

We propose a posteriori error estimators for first-order div least-squares (LS) finite element method for linear elasticity, Stokes equations and general second-order scalar elliptic problems. Our main interest is obtaining a posteriori error estimators for the dual variables (fluxes, strains, stress, etc.) which are main quantity of interest in many applications. We also provide a posteriori error estimators for the primary variable. These estimators are obtained from the local least-squares functional by assigning weight coefficients scaling the respective residuals. The weight coefficients are given in terms of local meshsize h_K. We establish the global upper bounds and local lower bounds for the estimators. The estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution (the indicators and estimators are identical). Numerical experiments show a superior performance of our a posteriori estimators for user-specific norm over the standard LS functional.     

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.     

A Posteriori Error Estimators for a Class of Variational Inequalities

In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system

We consider a conforming finite element approximation of the Reissner-Mindlin system. We propose a new robust a posteriori error estimator based on H(div) conforming finite elements and equilibrated fluxes. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests.     

Goal-oriented error estimation and adaptivity for the finite element method

In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Optimal non-linear estimation for distributed-parameter systems via the partition theorem

This paper considers the estimation problem for non-linear distributed-parameter systems via the ‘Partition Theorem’. First, the a posterioriprobability for the state is derived for the estimation of non-linear distributed-parameter systems. Secondly, linear systems excited by a white gaussian noise and with non-gaussian initial state are considered as a special class of the problem. The a posterioriprobability for the state, the optimal estimates and corresponding error covariance matrices are obtained by using the properties of the fundamental solution for the differential operator. Finally, it is shown that on approximate expression for the solution of the problem is also derived by applying a gaussian sum approximation technique.     

On the robustness of a multigrid method for anisotropic reaction-diffusion problems

In this paper, we consider a reaction-diffusion boundary value problem in a three-dimensional thin domain. The very different length scales in the geometry result in an anisotropy effect. Our study is motivated by a parabolic heat conduction problem in a thin foil leading to such anisotropic reaction-diffusion problems in each time step of an implicit time integration method [7]. The reaction-diffusion problem contains two important parameters, namely ε >0 which parameterizes the thickness of the domain and μ >0 denoting the measure for the size of the reaction term relative to that of the diffusion term. In this paper we analyze the convergence of a multigrid method with a robust (line) smoother. Both, for the W- and the V-cycle method we derive contraction number bounds smaller than one uniform with respect to the mesh size and the parameters ε and μ.      

A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem

In this paper we address several issues arising from a singularly perturbed fourth order problem with small parameter ε. First, we introduce a new family of non-conforming elements. We then prove that the corresponding finite element method is robust with respect to the parameter ε and uniformly convergent to order h ^1/2. In addition, we analyze the effect of treating the Neumann boundary condition weakly by Nitsche’s method. We show that such treatment is superior when the parameter ε is smaller than the mesh size h and obtain sharper error estimates. Such error analysis is not restricted to the proposed elements and can easily be carried out to other elements as long as the Neumann boundary condition is imposed weakly. Finally, we discuss the local error estimates and the pollution effect of the boundary layers in the interior of the domain.     

A space–time adaptation scheme for unsteady shallow water problems

We provide a space–time adaptation procedure for the approximation of the Shallow Water Equations (SWE). This approach relies on a recovery based estimator for the global discretization error, where the space and time contributions are kept separate. In particular we propose an ad hoc procedure for the recovery of the time derivative of the numerical solution and then we employ this reconstruction to define the error estimator in time. Concerning the space adaptation, we move from an anisotropic error estimator able to automatically identify the density, the shape and the orientation of the elements of the computational mesh. The proposed global error estimator turns out to share the good properties of each recovery based error estimator. The whole adaptive procedure is then combined with a suitable stabilized finite element SW solver. Finally the reliability of the coupled solution–adaptation procedure is successfully assessed on two unsteady test cases of interest for hydraulics applications.     

An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method Applied to Linear and Nonlinear Diffusion Problems

In this paper we present a new residual-based reliable a posteriori error estimator for the local discontinuous Galerkin approximations of linear and nonlinear diffusion problems in polygonal regions of R ². Our analysis, which applies to convex and nonconvex domains, is based on Helmholtz decompositions of the error and a suitable auxiliary polynomial function interpolating the Dirichlet datum. Several examples confirming the reliability of the estimator and providing numerical evidences for its efficiency are given. Furthermore, the associated adaptive method, which considers meshes with and without hanging nodes, is shown to be much more efficient than a uniform refinement to compute the discrete solutions. In particular, the experiments illustrate the ability of the adaptive algorithm to localize the singularities of each problem.Mathematics Subject Classifications (1991). 65N30This revised version was published online in July 2005 with corrected volume and issue numbers.     

A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen–Cahn equation u_t − Δu+ ε⁻² f(u)=0. It is shown that the error depends on ε⁻¹ only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen–Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.