Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.     

A posteriori error estimations of residual type for Signorini's problem

This paper presents an a posteriori error analysis for the linear finite element approximation of the Signorini problem in two space dimensions. A posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained. We perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error.     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

Duality Based A Posteriori Error Estimator for the Dirichlet Problem

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. All the derivations are done on continuous level, and the estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H¹, independently of the discretization method chosen. In particular, we make no use of the Galerkin orthogonality, which enables us to implement the estimator for measuring the error of the fictitious domain/penalty finite element method. The estimator is easily computable, and the only constant present in the estimator is the one from Friedrichs' inequality; the constant depends solely on the domain geometry, and the estimator is quite non‐sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.     

A posteriori error estimates for an optimal control problem of laser surface hardening of steel

The main focus of this article is on the development of a posteriori error estimates for an optimal control problem of laser surface hardening of steel, governed by a dynamical system consisting of a semi-linear parabolic equation and an ordinary differential equation. A posteriori error estimators are developed for the variables representing temperature, formation of austenite, and laser energy using residual method when a continuous piecewise linear discretization has been used for the finite element approximation of space variables and a discontinuous Galerkin method has been used for time and control discretizations. The error indicators are used in the implementation and numerical results are obtained.     

Error estimates for parabolic optimal control problems with control and state constraints

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.     

An adaptive finite element method for the sparse optimal control of fractional diffusion

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE

In this paper, we study adaptive finite element discretization schemes for an optimal control problem governed by elliptic PDE with an integral constraint for the state. We derive the equivalent a posteriori error estimator for the finite element approximation, which particularly suits adaptive multi-meshes to capture different singularities of the control and the state. Numerical examples are presented to demonstrate the efficiency of a posteriori error estimator and to confirm the theoretical results.     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

     

An adaptive least‐squares mixed finite element method for the Signorini problem

We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first‐order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in . This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest‐order Raviart‐Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two‐dimensional and three‐dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017     

A posteriori error analysis for a class of integral equations and variational inequalities

We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order ${2s \in (0,2]}We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ? (0,2]{2s \in (0,2]}. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H ^s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.     

具吸收项的退化抛物问题的有限元误差估计

1　引　　言本文将考虑下列退化抛物方程的Ｇａｌｅｒｋｉｎ逼近ｕｔ =Δβ(ｕ) - ｆ(ｕ)　　在Ω× ( 0 ,Ｔ]内 ( 1 .1 )ｕ(ｘ ,ｔ) =0　　　　　　　在 Ω× ( 0 ,Ｔ]上 ( 1 .2 )ｕ(ｘ ,0 ) =ｕ0 (ｘ)　　在Ω内(1 .3)其中Ω Ｒｎ 是有界凸域 ,0 <Ｔ <∞ .β(ｖ) (ｖ∈Ｒ)是满足 β( 0 ) =β′( 0 ) =0且 β′≥ 0的函数 因此 ,( 1 1 )是退化的非线性抛物方程 方程 ( 1 1 )具有深刻的物理背景[1] ,文献 [2 - 3]讨论了方程 ( 1 1 )的特殊形式—多孔介质方程 (ＰＭＥ)的数值方法 关于ＰＭＥ解的存在性、唯一性和正则性已有许多结果 …     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirich-let boundary conditions in a convex polygonal domain in the plane.This new class of finite elements,which is called composite finite elements,was first introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial differential equations on domains with complicated geometry.The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving par-tial differential equations by domain discretization method.The composite finite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the fine-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the finite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L∞(L2)-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.