A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM

In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.     

C^{0}-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem

In this paper, a theoretical framework is constructed on how to develop $C^0$ -nonconforming elements for the fourth order elliptic problem. By using the bubble functions, a simple practical method is presented to construct one tetrahedral $C^{0}$ -nonconforming element and two cuboid $C^{0}$ -nonconforming elements for the fourth order elliptic problem in three spacial dimensions. It is also proved that one element is of first order convergence and other two are of second order convergence. From the best knowledge of us, this is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.     

A new robust C-type nonconforming triangular element for singular perturbation problems

In this paper, a new robust C⁰ triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique, and a general convergence theorem for C⁰ nonconforming elements is presented. The convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter. Numerical experiments are also carried out to demonstrate the efficiency of the new element.     

A new high order compact off-step discretization for the system of 3D quasi-linear elliptic partial differential equations

We present a new fourth order compact finite difference scheme based on off-step discretization for the solution of the system of 3D quasi-linear elliptic partial differential equations subject to appropriate Dirichlet boundary conditions. We also develop new fourth order methods to obtain the numerical solution of first order normal derivatives of the solution. In all the cases, we use only 19-grid points of a single computational cell to compute the problem. The proposed methods are directly applicable to singular problems and the problems in polar coordinates, without any modification required unlike the previously developed high order schemes of [14] and [30]. We discuss the convergence analysis of the proposed method in details. Many physical problems are solved and comparative results are given to illustrate the usefulness of the proposed methods.     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A mixed finite element method for solving the nonstationary stokes equation

Summary This paper deals with a mixed finite element method for approximating a fourth order initial value problem arising from the nonstationary Stokes problem. For piecewise linear shape functions error estimates are given with convergence rates similar to the elliptic case. Some numerical computations will illustrate the theoretical results.     

A C0 interior penalty method for a singularly‐perturbed fourth‐order elliptic problem on a layer‐adapted mesh

We analyze the convergence of a continuous interior penalty (CIP) method for a singularly perturbed fourth‐order elliptic problem on a layer‐adapted mesh. On this anisotropic mesh, we prove under reasonable assumptions uniform convergence of almost order k ? 1 for finite elements of degree k ≥ 2. This result is of better order than the known robust result on standard meshes. A by‐product of our analysis is an analytic lower bound for the penalty of the symmetric CIP method. Finally, our convergence result is verified numerically. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 838–861, 2014     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

New adaptive mixed finite element method (AMFEM)

The need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality. We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods. Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second-Order Optimality Conditions for a Semilinear Elliptic Optimal Control Problem with Mixed Pointwise Constraints

This paper studies second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints. We show that in some cases, there is a common critical cone under which the second-order necessary and sufficient optimality conditions for the problem are valid. Our results approach to a theory of no-gap second-order conditions. In order to obtain such results, we reduce the problem to a special mathematical programming problem with polyhedricity constraint set. We then use some tools of variational analysis and techniques of semilinear elliptic equations to analyze second-order conditions.     

椭圆型方程组边值问题近似解的收敛性

本文研究了在矩形域内四阶系数不连续的椭圆型方程组边值问题近似解的收敛性问题,这对某一类弹性支承上的矩形板弯曲问题有参考价值。     

NON C~0 NONCONFORMING ELEMENTS FOR ELLIPTIC FOURTH ORDER SINGULAR PERTURBATION PROBLEM

In this paper we give a convergence theorem for non C^0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kind of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.     

Optimal control for elliptic systems with pointwise euclidean norm constraints on the controls

Optimal control for an elliptic system with pointwise Euclidean norm constraints on the control variables is investigated. First order optimality conditions are derived in a manner that is amenable for numerical realisation. An efficient semismooth Newton algorithm is proposed based on this optimality system. Numerical examples are given to validate the superlinear convergence of the semismooth Newton algorithm.     

Snapshot location for POD in control of a linear heat equation

In the present work, we study the approximation of a distributed optimal control problem for a linear heat equation with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). We show that snapshot location for control problems is crucial in model reduction. For the determination of the time instances (snapshot locations) we utilize an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present a numerical test to illustrate our approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter.     

求解椭圆边值问题惩罚形式的间断有限元方法

本文提出了一个新的求解二阶椭圆边值问题的惩罚形式间断有限元方法并给出了稳定性和收敛性分析. 特别地,本文建立了间断有限元解的基于余量的后验误差估计,给出了求解间断有限元方程的自适应算法.       

一个新的非常规各向异性元的收敛性分析

构造了一个新的非常规各向异性Hermite型矩形单元并据此对二阶椭圆问题提出了一个混合元格式,同时给出了该格式的收敛性分析.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

Discretization of elliptic control problems with time dependent parameters

We consider linear-quadratic problems of optimal control with an elliptic state equation and control constraints. For a discretization of the state equation by the method of Finite Differences and a piecewise approximation of the control we develop error estimates for the solution of the discrete problem and, based on the optimality conditions, we construct a new feasible control for which we derive error estimates of quadratic order.     

Numerical approximation of elliptic control problems with finitely many pointwise constraints

We study the numerical approximation of elliptic control problems with finitely many pointwise state constraints and control bounds. Results for the continuous problem are collected and a complete study of the discrete problems is carried out, including, existence of solutions, optimality conditions, convergence to solutions of the continuous problem and error estimates. A numerical example is provided.