A posteriori error estimates for adaptive finite element discretizations of boundary control problems (original) (raw)
Boundary Value Problems
In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in L 2-H 1 norms for the Neumann boundary optimal control problems governed by semilinear elliptic equations. We also give L 2-L 2 a posteriori error estimates for the optimal control problems. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximations for the semilinear Neumann boundary optimal control problems.
A review of some a posteriori error estimates for adaptive finite element methods
Mathematics and Computers in Simulation, 2010
Recently, the adaptive finite element methods have gained a very important position among numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. While the classical a posteriori error estimates are oriented to the use in h-methods the contemporary higher order hp-methods usually require new approaches in a posteriori error estimation. We present a brief review of some error estimation procedures for some particular both linear and nonlinear differential problems with special regards to the needs of the hp-method.
A posteriori error control for finite element approximations of elliptic eigenvalue problems
2001
We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.
An optimal control approach to a posteriori error estimation in finite element methods
Acta Numerica, 2001
This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the 'energy norm' or the L 2 norm, involving usually unknown 'stability constants'. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the 'dual-weighted-residual method', is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.
A Directional Error Estimator For Adaptive Finite Element Analysis
. We present an error estimator based on first- and second-order derivatives recovery for finite element adaptive analysis. At first, we briefly discuss the abstract framework of the adopted error estimation techniques. Some possibilities of derivatives recovery are considered, including the proposal of a directional error estimator. Using the directional error estimator proposed, an adaptive finite element analysis is performed which gives an adapted mesh where the estimated error is uniformly distributed over the domain. The advantages of adapting meshes are well known, but we place particular emphasis on the anisotropic mesh adaptation process generated by the directional error estimator. This mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori. We apply the above abstract formulation to analyze the behaviour of the recovery technique and the proposed adaptive process for some pa...
Dual boundary-element method: Simple error estimator and adaptivity
International Journal for Numerical Methods in Engineering, 2011
This paper is concerned with the effective numerical implementation of the adaptive dual boundary-element method (DBEM), for two-dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single-region analysis. Taking advantage on the use of non-conforming parametric boundary-elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM.
Error estimates for the numerical approximation of Neumann control problems
Computational Optimization and Applications, 2007
We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193-219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L 2 (). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45-61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided.
Adaptive Error Control for Multigrid Finite Element Methods
Computing, 1995
Adaptive Error Control for Multigrid Finite Element Methods. We consider the problem of adaptive error control in the finite element method including the error resulting from inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element method with a canonical multigrid algorithm. The proofs are based on a combination of so-called strong stability and the orthogonality inherent in both the finite element method and the multigfid algorithm.
A Posteriori Error Estimates of Residual Type for Second Order Quasi-Linear Elliptic PDEs
Journal of Mathematics Research, 2012
We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: −∇ • (α(x, ∇u)∇u) = f (x) in Ω ⊂ R 2 , u = 0 on ∂Ω, where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H 1-norm by an indicator η which is composed of L 2norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.