Gunther H. Peichl

Shape Optimization and Fictitious Domain Approach for Solving Free Boundary Problems of Bernoulli Type

This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems. It is based on a shape optimization approach. The state problems are solved by a fictitious domain solver using boundary Lagrange multipliers.

Estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique

SummaryIn this paper we apply a hybrid method to estimate a temporally and spatially varying diffusion coefficient in a parabolic system. This technique combines the output-least-squares- and the equation error method. The resulting optimization problem is solved by an augmented Lagrangian approach and convergence as well as rate of convergence proofs are provided. The stability of the estimated coefficient with respect to perturbations in the observation is guaranteed.

Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach

SUMMARY The paper deals with a fast method for solving large scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside of the original domain. This approach has a significantly higher convergence rate, however the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved by a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright c � 2007 John Wiley & Sons, Ltd.

The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley-Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples.

Numerical Gradients for Shape Optimization Based on Embedding Domain Techniques

An embedding domain technique is proposed to characterize the gradient of shape optimization problems. A discussion of the numerical realization of the arising saddle point problems is given and numerical feasibility of the gradient information is discussed.

An Immersed Interface Technique for the Numerical Solution of the Heat Equation on a Moving Domain

A finite difference scheme for the heat equation with mixed boundary conditions on a moving domain is presented. We use an immersed interface technique to discretize the Neumann condition and the Shortley–Weller approximation for the Dirichlet condition. Monotonicity of the discretized parabolic operator is established. Numerical results illustrate the feasibility of the approach.

Asymptotic analysis of stabilizability of a control system for a discretized boundary damped wave equation

This paper attempts to answer an open question on the uniform stabilizability of a commonly used finite difference method for the approximation of a control system modeled by 1-dimensional weakly damped wave equation. A detailed analysis of the spectral properties of the matrices used in the approximation and the asymptotic properties of the eigenvalues as a function of the dimension of the approximation space allow to conclude that finite dimensional control systems given by the approximation method considered here are not uniformly stabilizable. The results on the spectral properties of the matrices used in the finite difference approximation of the partial differential equations discussed in this paper also provide a method to calculate each individual eigenvalues of the matrix through a scalar iterative method. It is used in this paper to obtain accurate estimates on the stability and stabilizability margins of the finite dimensional control systems as well as to compute the eigenvalues numerically to...

Solving the Exterior Bernoulli Problem Using the Shape Derivative Approach

In this paper, we are interested in solving the exterior Bernoulli free boundary problem by minimizing a particular cost functional \(J\) over a class of admissible domains subject to two well-posed PDE constraints: a Dirichlet boundary value problem and a Neumann boundary value problem. The main result for this paper is the thorough computation of the first-order shape derivative of \(J\) using the shape derivatives of the state variables. At first, the material derivatives of the states are rigorously justified. Then the equation and the boundary conditions satisfied by the corresponding shape derivatives are derived directly from the definition of the shape derivative and the variational equation for the material derivative. It becomes apparent that the analysis of the shape derivatives of the states requires more regular domains. Finally, it is noted that the shape gradient agrees with the structure predicted by the Hadamard structure theorem.

On the First-Order Shape Derivative of the Kohn-Vogelius Cost Functional of the Bernoulli Problem

The exterior Bernoulli free boundary problem is being considered. The solution to the problem is studied via shape optimization techniques. The goal is to determine a domain having a specific regularity that gives a minimum value for the Kohn-Vogelius-type cost functional while simultaneously solving two PDE constraints: a pure Dirichlet boundary value problem and a Neumann boundary value problem. This paper focuses on the rigorous computation of the first-order shape derivative of the cost functional using the Holder continuity of the state variables and not the usual approach which uses the shape derivatives of states.

Control System Radii and Robustness Under Approximation

The purpose of this paper is twofold. First, we provide a short review and summarize results on the robustness of controllability and stabilizability for finite dimensional control problems. We discuss the computation of system radii which provide a measure of robustness. Second, we consider systems which arise as finite difference and finite element approximations to control systems defined by partial differential equations. In particular, we derive controllability criteria for approximations of the controlled heat equation which are easy to check numerically. For a particular example we establish tight theoretical upper and lower bounds on the controllability radii for the finite difference and finite element models and compare these bounds with numerical results. Finally, we present numerical results on stabilizability radii which suggests that conditioning of the LQR control problem may be measured by this radii.