Roland Becker - Academia.edu (original) (raw)
Papers by Roland Becker
Siam Journal on Control and Optimization, 2000
We develop a new approach towards error control and adaptivity for nite element discretizations i... more We develop a new approach towards error control and adaptivity for nite element discretizations in optimization problems governed by partial di erential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary optimality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. This approach facilitates control of the error with respect to any given quantity of physical interest. The speci c feature introduced by the optimization problem is the natural choice of the error-control functional to conincide with the cost functional of the optimization problem. In this case, the Lagrangian multiplier can directly be used in weighting the cell-residuals in the error estimator. This leads to a particularly simple and cost-e cient algorithm for adapting the mesh according to the particular needs of the optimization problem. This approach is developed and tested for simple model problems in optimal control of semiconductivity.
Acta Numerica, 2001
This article surveys a general approach to error control and adaptive mesh design in Galerkin fin... more 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.
Numerische Mathematik, 2004
In this paper we develop an a posteriori error estimator for parameter identification problems. T... more In this paper we develop an a posteriori error estimator for parameter identification problems. The state equation is given by a partial differential equation involving a finite number of unknown parameters. The presented error estimator aims to control the error in the parameters due to discretization by finite elements. For this, we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence. Exploiting the special structure of the parameter identification problem, allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm. Several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section. For the problems considered here, both, the efficiency of the estimator and the quality of the generated meshes are satisfactory.
Computer Methods in Applied Mechanics and Engineering, 2003
In this note, we propose and analyse a method for handling interfaces between nonmatching grids b... more In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
Mathematical Modelling and Numerical Analysis, 2003
In this note, we propose and analyse a method for handling interfaces between nonmatching grids b... more In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
Siam Journal on Control and Optimization, 2000
We develop a new approach towards error control and adaptivity for nite element discretizations i... more We develop a new approach towards error control and adaptivity for nite element discretizations in optimization problems governed by partial di erential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary optimality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. This approach facilitates control of the error with respect to any given quantity of physical interest. The speci c feature introduced by the optimization problem is the natural choice of the error-control functional to conincide with the cost functional of the optimization problem. In this case, the Lagrangian multiplier can directly be used in weighting the cell-residuals in the error estimator. This leads to a particularly simple and cost-e cient algorithm for adapting the mesh according to the particular needs of the optimization problem. This approach is developed and tested for simple model problems in optimal control of semiconductivity.
Acta Numerica, 2001
This article surveys a general approach to error control and adaptive mesh design in Galerkin fin... more 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.
Numerische Mathematik, 2004
In this paper we develop an a posteriori error estimator for parameter identification problems. T... more In this paper we develop an a posteriori error estimator for parameter identification problems. The state equation is given by a partial differential equation involving a finite number of unknown parameters. The presented error estimator aims to control the error in the parameters due to discretization by finite elements. For this, we consider the general setting of a partial differential equation written in weak form with abstract parameter dependence. Exploiting the special structure of the parameter identification problem, allows us to derive an error estimator which is cheap in comparison to the overall optimization algorithm. Several examples illustrating the behavior of an adaptive mesh refinement algorithm based on our error estimator are discussed in the numerical section. For the problems considered here, both, the efficiency of the estimator and the quality of the generated meshes are satisfactory.
Computer Methods in Applied Mechanics and Engineering, 2003
In this note, we propose and analyse a method for handling interfaces between nonmatching grids b... more In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
Mathematical Modelling and Numerical Analysis, 2003
In this note, we propose and analyse a method for handling interfaces between nonmatching grids b... more In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.