Christian Grossmann

Numerical treatment of partial differential equations

Contents Notation 1 Basics 1.1 Classification and Correctness 1.2 Fouriers Method, Integral Transforms 1.3 Maximum Principle, Fundamental Solution 2 Finite Difference Methods 2.1 Basic Concepts 2.2 Illustrative Examples 2.3 Transportation Problems and Conservation Laws 2.4 Elliptic Boundary Value Problems 2.5 Finite Volume Methods as Finite Difference Schemes 2.6 Parabolic Initial-Boundary Value Problems 2.7 Second-Order Hyperbolic Problems 3 Weak Solutions 3.1 Introduction 3.2 Adapted Function Spaces 3.3 VariationalEquationsand conformingApproximation 3.4 WeakeningV-ellipticity 3.5 NonlinearProblems 4 The Finite Element Method 4.1 A First Example 4.2 Finite-Element-Spaces 4.3 Practical Aspects of the Finite Element Method 4.4 Convergence of Conforming Methods 4.5 NonconformingFiniteElementMethods 4.6 Mixed Finite Elements 4.7 Error Estimators and adaptive FEM 4.8 The Discontinuous Galerkin Method 4.9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems 5.1 Parabolic Problems 5.2 Second-Order Hyperbolic Problems 6 Singularly Perturbed Boundary Value Problems 6.1 Two-Point Boundary Value Problems 6.2 Parabolic Problems, One-dimensional in Space 6.3 Convection-Diffusion Problems in Several Dimensions 7 Variational Inequalities, Optimal Control 7.1 Analytic Properties 7.2 Discretization of Variational Inequalities 7.3 Penalty Methods 7.4 Optimal Control of PDEs 8 Numerical Methods for Discretized Problems 8.1 Some Particular Properties of the Problems 8.2 Direct Methods 8.3 Classical Iterative Methods 8.4 The Conjugate Gradient Method 8.5Multigrid Methods 8.6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs Bibliography: Original Papers Index

Elliptic Control by Penalty Techniques with Control Reduction

The paper deals with the numerical treatment of optimal control problems with bounded distributed controls and elliptic state equations by a wider class of barrier-penalty methods. If the constraints are treated by barrier-penalty techniques then the necessary and sufficient optimality condition forms a coupled system of nonlinear equations which contain not only the usual adjoint and the state equation, but also an approximate projection by means of barrier-penalty terms. Under the made assumptions from the last one the control can be eliminated. This reduced optimality system which does not contain explicitly the controls, but the more regular states and adjoints only, is studied in detail.

On the solution of discretized obstacle problems by an adapted penalty method

In this paper we present a mutual adjustment for mesh size parameters of the discretization and for penalty parameters. This enables to restrict the error resulting from the penalty technique to the same order as the discretization error without destroying the conditioning of the problem. Furthermore we analyze the convergence of the discrete coincidence set.ZusammenfassungIn der vorliegenden Arbeit wird eine abgestimmte Wahl von Diskretisierungs- und Strafparametern zur Lösung diskretisierter Hindernisprobleme vorgeschlagen. Dabei gelingt es bei Erhaltung der Kondition des Problems, den Fehlereinfluß der Strafmethode auf die Ordnung des Diskretisierungsfehlers zu beschränken. Ferner wird die Konvergenz der diskretisierten Koinzidenzmengen analysiert.

Optimal Control of a Drying Process with Avoiding Cracks

The paper deals with the numerical treatment of the optimal con- trol of drying of materials which may lead to cracks. The drying process is controlled by temperature, velocity and humidity of the surrounding air. The state equations dene the humidity and temperature distribution within a sim- plied wood specimen for given controls. The elasticity equation describes the internal stresses under humidity and temperature changes. To avoid cracks these internal stresses have to be limited. The related constraints are treated by smoothed exact barrier-penalty techniques. The objective functional of the optimal control problem is of tracking type. Further it contains a quadratic regularization by an energy term for the control variables (surrounding air) and barrier-penalty terms. The necessary optimality conditions of the auxiliary problem form a coupled system of nonlinear equations in appropriate function spaces. This optimal- ity system is given by the state equations and the related adjoint equations, but also by an approximate projection onto the admissible set of controls by means of barrier-penalty terms. This system is discretized by nite elements and treated iteratively for given controls. The optimal control itself is per- formed by quasi-Newton techniques. 2000 AMS subject classications: 90C25, 90C51, 49K20, 65K10

Contraction behaviour of iteration–discretization based on gradient type projections

There is a wide range of iterative methods in infinite dimensional spaces to treat variational equations or variational inequalities. As a rule, computational handling of problems in infinite dimensional spaces requires some discretization. Any useful discretization of the original problem leads to families of problems over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration are embedded into each other. In the present paper, the behaviour of truncated iterative methods is studied, where at each discretization level only a finite number of steps is performed. In our study no accuracy dependent a posteriori stopping criterion is used. From an algorithmic point of view, the considered methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. A special emphasis is given on algorithms for the treatment of variational inequalities with strongly monotone operators over fixed point sets of quasi-nonexpansive mappings.

Mesh-independent convergence of penalty methods applied to optimal control with partial differential equations

In this paper, optimal control problems with elliptic state equations and constraints on controls are considered. Also state constraints are briefly discussed. Barrier-penalty methods are applied to treat the occurring restrictions. In the case of finite-dimensional optimization problems, the considered methods have a linear rate of convergence in dependence of the penalty parameter. However, in the case of infinite-dimensional problems, as studied in this article, the direct application of finite-dimensional theory, as given in Grossmann and Zadlo [A general class of penalty/barrier path-following Newton methods for nonlinear programming, Optimization 54 (2005), pp. 161–190], would lead to mesh-dependent order one estimates that deteriorate if the discretization is refined. In this article a first rigorous proof is given for inequality constrained problems that in the case of quadratic penalties a mesh-independence principle holds, i.e. the first-order convergence estimate holds for the continuous problem as well as for discretized problems independently of the discretization step size. The penalty techniques rest upon the control approximate reduction as discussed, e.g. in Grossmann et al. [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in System Modeling and Optimization, IFIP Advances in Information and Communication Technology, Vol. 312, Springer, Berlin, 2009, pp. 251–267; M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), pp. 45–61]. For the discretization conforming linear element discretization is applied. Some numerical examples illustrate and confirm the theoretical results.

A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control

The quadratic loss penalty is a well known technique for opti- mization and control problems to treat constraints. In the present paper they are applied to handle control bounds in a boundary control problems with semilinear elliptic state equations. Unlike in the case of finite dimensional optimization for infinite dimensional problems the order of convergence could only be roughly estimated, but numerical experiments revealed a clearly better convergence behavior with constants independent of the dimension of the used discretization. The main result in the present paper is the proof of sharp con- vergence bounds for both, the finite und infinite dimensional problem with a mesh-independence in case of the discretization. Further, to achieve an efficient realization of penalty methods the principle of control reduction is applied, i.e. the control variable is represented by the adjoint state variable by means of some nonlinear function. The resulting optimality system this way depends only on the state and adjoint state. This system is discretized by conforming linear finite elements. Numerical experiments show exactly the theoretically predicted behavior of the studied penalty technique.

Layer-adapted methods for a singularly perturbed singular problem

Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.

A direct higher order discretization in singular perturbations via domain split: A computational approach

In this paper the domain split concept as known from domain decomposition is combined with higher order difference discretizations to construct direct methods for the numerical treatment of singularly perturbed two-point boundary value problems.

Differentiable Local Barrier-Penalty Paths

Perturbations of Karush-Kuhn-Tucker conditions play an important role for primal-dual interior point methods. Beside the usual logarithmic barrier various further techniques of sequential unconstrained minimization are well known. However other than logarithmic embeddings are rarely studied in connection with Newton path-following methods. A key property that allows to extend the class of methods is the existence of a locally Lipschitz continuous path leading to a primal-dual solution of the KKT-system. In this paper a rather general class of barrier/penalty functions is studied. In particular, under LICQ regularity and strict complementarity assumptions the differentiability of the path generated by any choice of barrier/penalty functions from this class is shown. This way equality as well as inequality constraints can be treated directly without additional transformations. Further, it will be sketched how local convergence of the related Newton path-following methods can be proved without direct applications of self-concordance properties.