Adaptive Finite Element Approximation Research Articles

Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

AbstractIn this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

SUMMARYAn adaptive finite element approximation for an optimal control problem of the Stokes flow with an L2‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H1‐error of state and L2‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.

Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

AbstractIn this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one‐way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem. Copyright © 2010 John Wiley & Sons, Ltd.

An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture

The energy of the Francfort-Marigo model of brittle fracture can be approximated, in the sense of $\Gamma$-convergence, by the Ambrosio-Tortorelli functional. In this work, we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm, we combine a Newton-type method with residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio-Tortorelli functional.

Local Multilevel Methods for Adaptive Finite Element Methods for Nonsymmetric and Indefinite Elliptic Boundary Value Problems

In this paper, we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems. Two types of local smoothers are constructed. One is based on the original nonsymmetric problems, and the other is defined in terms of the associated symmetric problems. It is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal, which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine. Numerical experiments are reported to confirm our theory.

Adaptive finite element approximations on nonmatching grids for second‐order elliptic problems

AbstractIn this article we consider a finite element approximation for a model elliptic problem of second order on non‐matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.

Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer

AbstractIn this paper we develop an adaptive finite element method for heat transfer in incompressible fluid flow. The adaptive method is based on an a posteriori error estimate for the coupled problem, which identifies how accurately the flow and heat transfer problems must be solved in order to achieve overall accuracy in a specified goal quantity. The a posteriori error estimate is derived using duality techniques and is of dual weighted residual type. We consider, in particular, an a posteriori error estimate for a variational approximation of the integrated heat flux through the boundary of a hot object immersed into a cooling fluid flow. We illustrate the method on some test cases involving three‐dimensional time‐dependent flow and transport. Copyright © 2008 John Wiley & Sons, Ltd.

Adaptive finite element approximation of multiphysics problems

AbstractSimulation of multiphysics problems is a common task in applied research and industry. Often a multiphysics solver is built by connecting several single‐physics solvers into a network. In this paper, we develop a basic adaptive methodology for such multiphysics solvers. The adaptive methodology is based on a posteriori error estimates that capture the influence of the discretization errors in the different solvers on a given functional output. These estimates are derived using duality‐based techniques. Copyright © 2007 John Wiley & Sons, Ltd.

Ana posteriorierror analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity

In our earlier work [E. Stein, S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method, Comput. Methods Appl. Mech. Engrg. 150 (1997) 327–350; E. Stein, S. Ohnimus, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg. 176 (1999) 363–385; E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg. 60 (2004) 103–138.], residual-based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of Neumann problems on the element level. In particular, these estimators control the errors obtained while expanding the model hierarchically from 1D- to 2D- and from 2D- to 3D-elastic theories in the (global) energy norm. In this paper, we extend both types of estimates to goal-oriented a posteriori error estimation within the framework of nonlinear structural mechanics. Structural model adaptivity is characterized by the fact that the dimensions, in general the constitutive equations and especially the related subdomains—and thus the test and solution spaces of the process-dependent models considered—are different. This has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model “ m” to those of the refined model “ m + 1” are necessary. Therefore, such prolongation operators are discussed in detail. Furthermore, duality techniques are used for the model and discretization error estimates, using system and load-dependent quantities of interest. In this paper, we also consider different finite element meshes for the primal and the dual problem. Finally, some numerical examples demonstrate the expected effects, realizing verification and even validation in structural mechanics, i.e. approaching a new paradigm in Computational Mechanics for achieving reliability and efficiency.

The Finite Element Approximation of the Nonlinear Poisson–Boltzmann Equation

A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson–Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.

A posteriori error estimates for adaptive finite element discretizations of boundary control problems

We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the adjoint state, the control, and the adjoint control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations. The proof of the efficiency and reliability is done without any regularity assumption. Adaptive mesh refinement is realized on the basis of a bulk criterion. The performance of the adaptive finite element approximation is illustrated by a detailed documentation of numerical results for selected test problems.

A posteriori error estimates for control problems governed by nonlinear elliptic equations

In this paper, we derive a posteriori error estimates for the finite element approximation of a class of nonlinear optimal control problems. We derive a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for the control problems.

An iterative method for adaptive finite element solutions of an energy transport model of semiconductor devices

A self-adjoint formulation of the energy transport model of semiconductor devices is proposed. This new formulation leads to symmetric and monotonic properties of the resulting system of nonlinear algebraic equations from an adaptive finite element approximation of the model. A node-by-node iterative method is then presented for solving the system. This is a globally convergent method that does not require the assembly of the global matrix system and full Jacobian matrices. An adaptive algorithm implementing this method is described in detail to illustrate the main features of this paper, namely, adaptation, node-by-node calculation, and global convergence. Numerical results of simulations on deep-submicron diode and MOSFET device structures are given to demonstrate the accuracy and efficiency of the algorithm.

A posteriori error estimates for optimal control problems governed by parabolic equations

In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem.

A Newton–Krylov solver with multiplicative Schwarz preconditioning for finite element compressible flow simulations

AbstractThe paper presents an implementation of the Newton–Krylov–Schwarz methodology for stabilized, adaptive finite element approximations of compressible Navier–Stokes equations. Special emphasis is laid on preconditioning of the GMRES method within Newton iterations. Several Schwarz preconditioners are compared on a benchmark problem, including a new implementation of multiplicative preconditioner for matrix‐free GMRES, that can be interpreted as accelerated block Gauss–Seidel iterations. Numerical examples show a good performance of the new algorithm, for single step problems and overall simulations. Copyright © 2002 John Wiley & Sons, Ltd.

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.

A Posteriori Error Estimates for Control Problems Governed by Stokes Equations

In this paper, we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the Stokes equations. We obtain a posteriori error estimators for both the state and the control approximation in the L2 norm and the H1 norm. These estimates can be used to construct reliable adaptive finite element approximation for the control problems.