In this paper we derive a posteriori error estimates for space-time finite element discretizations of parabolic optimization problems. The provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization (time, space, and control discretization). This allows us to set up an efficient adaptive algorithm which successively improves the accuracy of the computed solution by construction of locally refined meshes for time and space discretizations.

/pdf/adaptive-space-time-finite-element-methods-for-parabolic-d0dcevkmwc.pdf

Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems

In this paper, a priori error estimates for space–time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived. We extend the approach from Meidner and Vexler (SIAM Control Optim 47(3):1150–1177, 2008; SIAM Control Optim 47(3):1301–1329, 2008) where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constant functions in time and cellwise constant functions in space are derived in detail and we explain how error estimate for further discretization approaches, e.g., cellwise linear discretization in space, the postprocessing approach from Meyer and Rosch (SIAM J Control Optim 43:970–985, 2004), and the variationally discrete approach from Hinze (J Comput Optim Appl 30:45–63, 2005) can be obtained. In addition, we derive an estimate for a setting with finitely many time-dependent controls.

A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems

A priori error estimates for elliptic optimal control problems with a bilinear state equation

We present an efficient and easy to implement approach to solving the semidiscrete equation systems resulting from time discretization of nonlinear parabolic problems with discontinuous Galerkin methods of order $$r$$ . It is based on applying Newton's method and decoupling the Newton update equation, which consists of a coupled system of $$r+1$$ elliptic problems. In order to avoid complex coefficients which arise inevitably in the equations obtained by a direct decoupling, we decouple not the exact Newton update equation but a suitable approximation. The resulting solution scheme is shown to possess fast linear convergence and consists of several steps with same structure as implicit Euler steps. We construct concrete realizations for order one to three and give numerical evidence that the required computing time is reduced significantly compared to assembling and solving the complete coupled system by Newton's method.

Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems

In this article we summarize recent results on a priori error estimates for space-time finite element discretizations of linear-quadratic parabolic optimal control problems. We consider the following three cases: problems without inequality constraints, problems with pointwise control constraints, and problems with state constraints pointwise in time. For all cases, error estimates with respect to the temporal and to the spatial discretization parameters are derived. The results are illustrated by numerical examples.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems

This paper is the second part of our work on a priori error analysis for finite element discretizations of parabolic optimal control problems. In the first part [SIAM J. Control Optim., 47 (2008), ...

VexlerBoris

Papers

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part II: Problems with Control Constraints