A discontinuous Galerkin method for the Rosenau equation

A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

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 discontinuous Galerkin method for the Cahn-Hilliard equation

The Cahn-Hilliard equation is modeled to describe the dynamics of phase separation in glass and polymer systems. A priori error estimates for the Cahn-Hilliard equation have been studied by the authors. In order to control accuracy of approximate solutions, a posteriori error estimation of the Cahn-Hilliard equation is obtained by discontinuous Galerkin method.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Efficiency of a posteriori BEM--error estimates for first-kind integral equations on quasi--uniform meshes

In the numerical treatment of integral equations of the first kind using boundary element methods (BEM), the author and E. P. Stephan have derived a posteriori error estimates as tools for both reliable computation and self-adaptive mesh refinement. So far, efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only. This work affirms efficiency by proving the reverse inequality. Based on best approximation, on inverse inequalities and on stability of the discretization, and complementary to our previous work, an abstract approach yields a converse estimate. This estimate proves efficiency of an a posteriori error estimate in the BEM on quasi--uniform meshes for Symm's integral equation, for a hypersingular equation, and for a transmission problem.

     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

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.     

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.     

A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation

In this paper, we study an edge-stabilization Galerkin approximation scheme for the constrained optimal-control problem governed by convection-dominated diffusion equation. The method uses least-square stabilization of the gradient jumps across element edges. A priori and a posteriori error estimates are derived for both the state, co-state and the control. The theoretical results are illustrated by two numerical experiments.     

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L^∞(L²) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L^∞(L²) norm.     

A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation

An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is presented such that it can be used for adaptive strategies based on dual weighted residual methods. A posteriori error estimates based on weighted global projections and local projections are also proved.     

Numerical approximation and error control for a thermoelastic contact problem

A numerical method using finite elements for the spatial discretization and the Crank–Nicolson scheme for the time stepping is applied to a partial differential equation problem involving thermoelastic contact. The Crank–Nicolson scheme is interpreted as a low order continuous Galerkin method. By exploiting the variational framework inherent in this approach, an a posteriori error estimate is derived. This estimate gives a bound on the approximation error that depends on computable quantities such as the mesh parameters, time step and numerical solution. In this paper, the a posteriori estimate is used to develop a time step refinement strategy. Several computational examples are included that demonstrate the performance of the method and validity of the estimate.     

Dual weighted residual method for optimal control of hyperbolic equations of second order

In this paper the dual weighted residual method for optimal control problems of hyperbolic equations of second order is considered. The state equation is written as a first order system in time and a posteriori error estimates separating the influences of time, space, and control discretization are derived to obtain a better accurancy of the discrete solution. A numerical example for optimal control of a nonlinear wave equation is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

Summary. 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. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

A posteriori error estimates for optimal distributed control governed by the evolution equations

We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the L²-norm in the space-time cylinder combined with a special time weighted energy norm.     

Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints

In this paper, we study a posteriori error estimates of the upwind symmetric interior penalty Galerkin (SIPG) method for the control constrained optimal control problems governed by linear diffusion–convection–reaction partial differential equations. Residual based error estimators are used for the state, the adjoint and the control. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented for convection dominated problems to illustrate the theoretical findings and the effectiveness of the adaptivity.     

GUARANTEED BOUNDS FOR THE SOLUTION OF THE WAVE EQUATION

Abstract

Numerical solution of the wave equation in the form of close lower and upper bounds provides a secure a posteriori error estimate that can be used for efficient accuracy control. The method considered in this paper uses some monotone properties of the differential operator in the wave equation to construct bounds for the solution in the form of trigonometric polynomials of x. Aspects of the numerical implementation, the accuracy of the computed bounds and some numerical examples are discussed.     

A posteriori error estimates for the effective Hamiltonian of dislocation dynamics

We study an implicit and discontinuous scheme for a non-local Hamilton?CJacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall?CLions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided.     

奇异摄动反应扩散方程的后验误差估计及自适应算法

研究了一类奇异摄动半线性反应扩散方程的自适应网格方法.在任意非均匀网格上建立迎风有限差分离散格式,并推导出离散格式的后验误差界,然后用该误差界设计自适应网格移动算法.数值实验结果证明了所提出的自适应网格方法的有效性.