Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

In this paper, we investigate the superconvergence of fully discrete splitting positive definite mixed finite element (MFE) methods for parabolic optimal control problems. For the space discretization, the state and co-state are approximated by the lowest order Raviart–Thomas MFE spaces and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. We derive the superconvergence between the projections of exact solutions and numerical solutions or the exact solutions and postprocessing numerical solutions for the control, state and co-state. A numerical example is provided to validate the theoretical results.     

Nonnegative control of finite-dimensional linear systems

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

Numerical solution of the heat equation with nonlinear boundary conditions in unbounded domains

The numerical solution of the heat equation in unbounded domains (for a 1D problem‐semi‐infinite line and for a 2D one semi‐infinite strip) is considered. The artificial boundaries are introduced and the exact artificial boundary conditions are derived. The original problems are transformed into problems on finite domains. The space semi‐discretization by finite element method and the full approximation by the implicit‐explicit Euler's method are presented. The solvability of the full discretization schemes is analyzed. Computational examples demonstrate the accuracy and the efficiency of the algorithms. Also, the behavior of blowing up solutions is examined numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 379–399, 2007     

A numerical analysis of the Cahn–Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.     

A numerical method based on extended Raviart–Thomas (ER‐T) mixed finite element method for solving damped Boussinesq equation

In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Simulation of Rayleigh waves in cracked plates

The aim of this paper is to develop new numerical procedures to detect micro cracks, or superficial imperfections, in thin plates using excitation by Rayleigh waves. We shall consider a unilateral contact problem between the two sides of the crack in an elastic plate subjected to suitable boundary conditions in order to reproduce a single Rayleigh wave cycle. An approximate solution of this problem will be calculated by using one of the Newmark methods for time discretization and a finite element method for space discretization. To deal with the nonlinearity due to the contact condition, an iterative algorithm involving one multiplier will be used; this multiplier will be approximated by using Newton's techniques. Finally, we will show numerical simulations for both cracked and non‐cracked plates. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical Solution of Hamilton-Jacobi-Bellman Equations by an Upwind Finite Volume Method

In this paper we present a finite volume method for solving Hamilton-Jacobi-Bellman(HJB) equations governing a class of optimal feedback control problems. This method is based on a finite volume discretization in state space coupled with an upwind finite difference technique, and on an implicit backward Euler finite differencing in time, which is absolutely stable. It is shown that the system matrix of the resulting discrete equation is an M-matrix. To show the effectiveness of this approach, numerical experiments on test problems with up to three states and two control variables were performed. The numerical results show that the method yields accurate approximate solutions to both the control and the state variables.     

A mixed collocation–finite difference method for 3D microscopic heat transport problems

Three-dimensional time-dependent initial-boundary value problems of a novel microscopic heat equation are solved by the mixed collocation–finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width. The collocation method on fixed grid size is used to approximate the space operator, whereas the finite difference scheme is used for time discretization. This new mixed method is applied to a novel heat problem in a particle, in order to compute the temperature distribution in and on the particle's surface. The second derivatives of the basis functions for the spectral approximation are derived. Direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method. The high-order accuracy and resolution achieved by the proposed method allows one to obtain engineering-accuracy solution on coarse meshes. The consistency, stability and convergence analysis are provided and numerical results are presented.     

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.     

Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations

This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

Uniform controllability properties for space/time-discretized parabolic equations

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ-schemes with \({\theta \in [\frac{1}2,1]}\) . For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions’ Hilbert Uniqueness Method strategy, exploiting the relaxed uniform observability estimate. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove an error bound between the fully discrete and the semi-discrete control functions. This bound is however not uniform with respect to the space discretization. The theoretical results are illustrated through numerical experimentations.     

On the cibstryctuib of discretizations of elliptic partial differential equations

Algorithmic aspects of a class of finite element collocation methods for the approximate numerical solution of elliptic partial differential equations are described Locall for each finite element the approximate solution is a polynomial. polynomials corresponding toadjacent finite elements need not match continuously but their values and noumal derivatives match at a discrete set of points on the common boundary.High order accuracy can be attained by increasing the number of mathching points and the number of colloction points for each finite element.Forlinear equations the collocation methods can be equivalently definde as generlized finite difference methods. The linear (or linearzed )equations that arise from the discretization lend themselves well to solution by the methods of the methods nested dissection.An implememtation is described and some numerical results are givevn.     

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—Strong convergence of optimal controls

A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.Dedicated to the 60th birthday of Lothar von Wolfersdorf     

Adaptive multilevel methods in space and time for parabolic problems-the periodic case

The aim of this paper is to display numerical results that show the interest of some multilevel methods for problems of parabolic type. These schemes are based on multilevel spatial splittings and the use of different time steps for the various spatial components. The spatial discretization we investigate is of spectral Fourier type, so the approximate solution naturally splits into the sum of a low frequency component and a high frequency one. The time discretization is of implicit/explicit Euler type for each spatial component. Based on a posteriori estimates, we introduce adaptive one-level and multilevel algorithms. Two problems are considered: the heat equation and a nonlinear problem. Numerical experiments are conducted for both problems using the one-level and the multilevel algorithms. The multilevel method is up to 70% faster than the one-level method.

     

Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: Computational Approach

This article is concerned with the numerical solution of multiobjective control problems associated with nonlinear partial differential equations and more precisely the Burgers equation. For this kind of problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. To compute the solution of the problem, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and a quasi-Newton BFGS algorithm for the iterative solution of the discrete control problem. Finally, we apply the above methodology to the solution of several tests problems. To be able to compare our results with existing results in the literature, we discuss first a single-objective control problem, already investigated by other authors. Finally, we discuss the multiobjective case.     

On box schemes for elartliptic variational inequalities

General finite volume approximations in two and three space dimensions are studied for the discretization of interior and boundary obstacle problems with mixed boundary conditions. First order convergence of the finite volume (box) solution is shown in the energy norm. Based on a discrete maximum principle there are proposed two penalization techniques for the solution of the finite volume inequalities. By coupling of discretization and penalty parameters the overall error is analyzed. The iterative solution of the penalty problems is discussed. Finally, results of numerical experiments are presented to illustrate the convergence behaviour between the exact, the box and the penalty solutions.     

一类带有铁磁材料参数的非线性涡流问题的A-φ有限元法

针对带有铁磁材料的非线性涡流问题,其非线性性通常体现在磁场强度和磁感应强度的关系上.本文提出了一种全离散的有限元A-φ格式,分别在时间和空间上采用向后欧拉公式以及节点有限元进行离散.首先,在合适的函数空间里给出时间上的半离散格式,通过考察其弱形式建立相应的适定性理论,并证明近似解收敛于弱解.其次,给出全离散格式并讨论其误差估计.最后,给出两个数值算例以验证理论结果.