A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

In this paper, we derive a posteriori error estimates of recovery type, and present the superconvergence analysis for the finite element approximation of distributed convex optimal control problems. We provide a posteriori error estimates of recovery type for both the control and the state approximation, which are generally equivalent. Under some stronger assumptions, they are further shown to be asymptotically exact. Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as a reliability bound for the control problems. Numerical results demonstrating our theoretical results are also presented in this paper.     

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.     

An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes

A posteriori error estimates for two-body contact problems are established. The discretization is based on mortar finite elements with dual Lagrange multipliers. To define locally the error estimator, Arnold–Winther elements for the stress and equilibrated fluxes for the surface traction are used. Using the Lagrange multiplier on the contact zone as Neumann boundary conditions, equilibrated fluxes can be locally computed. In terms of these fluxes, we define on each element a symmetric and globally H(div)-conforming approximation for the stress. Upper and lower bounds for the discretization error in the energy norm are provided. In contrast to many other approaches, the constant in the upper bound is, up to higher order terms, equal to one. Numerical examples illustrate the reliability and efficiency of the estimator. This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, B8.     

A Priori Error Estimates for Optimal Control Problems Governed by Transient Advection-Diffusion Equations

In this paper, we investigate a characteristic finite element approximation of quadratic optimal control problems governed by linear advection-dominated diffusion equations, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. We derive some a priori error estimates for both the control and state approximations. It is proved that these approximations have convergence order , where h _U and h are the spatial mesh-sizes for the control and state discretization, respectively, and k is the time increment. Numerical experiments are presented, which verify the theoretical results. This research was supported by the National Basic Research Program of China (No. 2007CB814906) and the National Natural Science Foundation of China (No. 10771124).     

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming Crouzeix-Raviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.     

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.     

An A Posteriori Error Estimate for the Local Discontinuous Galerkin Method

In this work an a posteriori global error estimate for the Local Discontinuous Galerkin (LDG) applied to a linear second order elliptic problem is analyzed. Using a mixed formulation, an upper bound of the error in the primal variable is derived from explicit computations. Finally, a local adaptive scheme based on explicit error estimators is studied numerically using one dimensional problems.This revised version was published online in July 2005 with corrected volume and issue numbers.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

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.     

Cost analysis of the longest-side (triangle bisection) refinement algorithm for triangulations

The triangulation refinement problem, as formulated in the adaptive finite element setting (also useful in the rendering of complex scenes), is discussed. This can be formulated as follows: given a valid, non-degenerate triangulation of a polygonal region, construct a locally refined triangulation, with triangles of prescribed size in a refinement regionR, and such that the smallest (or the largest) angle is bounded. To cope with this problem, longest-side refinement algorithms guarantee the construction of good quality irregular triangulations. This is due in part to their natural refinement propagation strategy farther than the (refinement) area of interestR. In this paper we prove that, asymptotically, the numberN of points inserted inR to obtain triangles of prescribed size, is optimal. Furthermore, in spite of the unavoidable propagation outside the refinement regionR, the time cost of the algorithm is linear inN, independent of the size of the triangulation. Specifically, the number of points inserted outsideR is of orderO(n log ₂ n) whereN=O(n²). We prove the latter result for circular and rectangular refinement regions, which allows us to conclude that this is true for general convex refinement regions. We also include empirical evidence, both in two and three dimensions, which is in complete agreement with the theory, even for small values ofN.     

A solution-adaptive central-constraint transport scheme for magnetohydrodynamics

The central-constraint transport scheme for magnetohydrodynamics in Ziegler [J. Comput. Phys. 196 (2004) 393] is made adaptive employing a block-structured mesh refinement method. Based on the guidelines in Berger and Collela [J. Comput. Phys. 82 (1989) 64] a mesh refinement variant has been developed which combines a flexible grid adaptation by using small blocks as refinement elements with integration speed by reducing the inherent overhead via block clustering techniques. The algorithms are discussed in detail and the efficiency of the implementation is benchmarked in terms of an efficiency parameter which takes into account both the obtained speedup factor and an error estimate. The three-dimensional benchmark problems are a spherical implosion problem and a shock-cloud collision problem in a magnetic medium. Further examples of astrophysical interest are presented which demonstrate the robustness and versatility of the new adaptive grid code.     

Bivariate Product Cubature Using Peano Kernels for Local Error Estimates

The error estimates of automatic integration by pure floating-point arithmetic are intrinsically embedded with uncertainty. This in critical cases can make the computation problematic. To avoid the problem, we use product rules to implement a self-validating subroutine for bivariate cubature over rectangular regions. Different from previous self-validating integrators for multiple variables (Storck in Scientific Computing with Automatic Result Verification, pp. 187–224, Academic Press, San Diego, [1993]; Wolfe in Appl. Math. Comput. 96:145–159, [1998]), which use derivatives of specific higher orders for the error estimates, we extend the ideas for univariate quadrature investigated in (Chen in Computing 78(1):81–99, [2006]) to our bivariate cubature to enable locally adaptive error estimates by full utilization of Peano kernels theorem. The mechanism for active recognition of unreachable error bounds is also set up. We demonstrate the effectiveness of our approach by comparing it with a conventional integrator.     

Weighted Error Estimates for Finite Element Solutions of Variational Inequalities

In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes. Received September 6, 1999; revised February 8, 2000     

Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery

In this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented.     

Adaptive mesh refinement using piecewise-linear shape functions based on the blending function method

Use of quadrilateral elements for finite element mesh refinement can lead either to so-called irregular meshes or the necessity of adjustments between finer and coarser parts of the mesh necessary. In the case of irregular meshes, constraints have to be introduced in order to maintain continuity of the displacements. Introduction of finite elements based on blending function interpolation shape functions using piecewise boundary interpolation avoids these problems. This paper introduces an adaptive refinement procedure for these types of elements. The refinement is anh-method. Error estimation is performed using the Zienkiewicz-Zhu method. The refinement is controlled by a switching function representation. The method is applied to the plane stress problem. Numerical examples are given to show the efficiency of the methodology.     

Adaptive mixed finite element method for Reissner–Mindlin plate

Mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of Reissner–Mindlin plate models. Automatic adaptive mesh-refining algorithms are an important tool to improve the approximation behavior of the finite element discretization. In this paper, a reliable and robust residual-based a posteriori error estimate is derived, which evaluates a t-depending residual norm based on results in [D. Arnold, R. Falk, R. Winther, Math. Modell. Numer. Anal. 31 (1997) 517–557]. The localized error indicators suggest an adaptive algorithm for automatic mesh refinement. Numerical examples prove that the new scheme is efficient.     

Error Estimates of a FEM with Lumping for Parabolic PDEs

L ²-norm. This well-known FEM is given by the use of the vertical line method and conforming linear finite elements on a triangulation. The main result of the paper are new estimates in the L ²-norm for the additional error term originated by lumping. Using these ones, for the FEM with lumping we can apply directly the proof technique of error estimates known for conforming FEMs. Received May 17, 2001; revised November 2, 2001 Published online February 18, 2002     

Equivalent a posteriori error estimates for elliptic optimal control problems with L1-control cost

An elliptic optimal control problem involving the $L^1$ norm of the control in the cost functional is considered in this paper. We use the full discretization and the variational discretization to approximate the control problem and the efficient and reliable a posteriori error estimates are obtained for the two cases. For the variational discretization, we also analyze the convergence of adaptive finite element methods. In the end, some examples are provided to validate our analysis.