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.     

Discontinuous Galerkin error estimation for hyperbolic problems on unstructured triangular meshes

We extend the error analysis of Adjerid and Baccouch [1], [2] for the discontinuous Galerkin discretization error to variable-coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes. We further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in [1] still hold for both steady and transient variable-coefficient linear and nonlinear problems. This local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes. We illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples.     

An a posteriori error estimator for model adaptivity in electrocardiology

We introduce an a posteriori modeling error estimator for the effective computation of electric potential propagation in the heart. Starting from the Bidomain problem and an extended formulation of the simplified Monodomain system, we build a hybrid model, called Hybridomain, which is dynamically adapted to be either Bi- or Monodomain ones in different regions of the computational domain according to the error estimator. We show that accurate results can be obtained with the adaptive Hybridomain model with a reduced computational cost compared to the full Bidomain model. We discuss the effectivity of the estimator and the reliability of the results on simulations performed on real human left ventricle geometries retrieved from healthy subjects.     

A Posteriori Error Estimation for a Finite Volume Discretization on Anisotropic Meshes

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.     

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.     

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.     

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.     

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.     

Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach–Ne?as–Babu?ka theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler–Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to \((k+1)\) -degree Radau polynomials, when polynomials of total degree not exceeding \(k\) are used. These results allow us to prove that the \(k\) -degree LDG solution and its derivatives are \(\mathcal O (h^{k+3/2})\) superconvergent at the roots of \((k+1)\) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time \(t\) converge to the true errors at \(\mathcal O (h^{k+5/4})\) rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the \(L^2\) -norm converge to unity at \(\mathcal O (h^{1/2})\) rate. Our proofs are valid for arbitrary regular meshes and for \(P^k\) polynomials with \(k\ge 1\) , and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.     

Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N, typically very small, and the (approximate) parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.In our earlier work, we develop a rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not complete. In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and model truncation effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.     

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a Gårding type inequality. Optimal order L ₂ norm a priori error estimates are derived for an adjoint consistent interior penalty method.     

Inhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem

We study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous Dirichlet boundary condition. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive a posteriori error bounds which are in simpler form and efficient. To accomplish this, we construct and use a post-processed solution ${\stackrel{?}{u}}_h$ of the discrete solution $u_h$ which satisfies the exact boundary conditions sharply although the discrete solution $u_h$ may not satisfy. We propose two post processing methods and analyze them, namely the harmonic extension and a linear extension. The theoretical results are illustrated by the numerical results.     

A posteriori error estimation by postprocessor independent of method of flowfield calculation

We consider a postprocessor that is able to analyze the flow-field generated by an external (unknown) code so as to determine the error of useful functionals. The residuals engendered by the action of a high-order finite-difference stencil on a numerically computed flow-field are used for adjoint based a posteriori error estimation. The method requires information on the physical model (PDE system), flowfield parameters and corresponding grid and may be constructed without availability of detailed information on the numerical method used for the flow computation.     

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 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     

Space–time adaptive finite difference method for European multi-asset options

The multi-dimensional Black–Scholes equation is solved numerically for a European call basket option using a priori–a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions.     

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.