Janet S. Peterson

Analysis and approximation of the Ginzburg-Landau model of superconductivity

The authors consider the Ginzburg–Landau model for superconductivity. First some well-known features of superconducting materials are reviewed and then various results concerning the model, the resultant differential equations, and their solution on bounded domains are derived. Then, finite element approximations of the solutions of the Ginzburg–Landau equations are considered and error estimates of optimal order are derived.

The reduced basis method for incompressible viscous flow calculations

The reduced basis method is a type of reduction method that can be used to solve large systems of nonlinear equations involving a parameter. In this work, the method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier–Stokes equations by finite–element methods. This paper demonstrates that the reduced basis method can be implemented to approximate efficiently solutions to incompressible viscous flows. Choices of basis vectors, issues concerning the implementation of the method, and numerical calculations are discussed. Two fluid flow calculations are considered, the driven cavity problem and flow over a forward facing step.

On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics

The authors consider the equations of stationary, incompressible magneto-hydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. They discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

Comparison of pure and Latinized centroidal Voronoi tessellation against various other statistical sampling methods

Abstract A recently developed centroidal Voronoi tessellation (CVT) sampling method is investigated here to assess its suitability for use in statistical sampling applications. CVT efficiently generates a highly uniform distribution of sample points over arbitrarily shaped M -dimensional parameter spaces. On several 2-D test problems CVT has recently been found to provide exceedingly effective and efficient point distributions for response surface generation. Additionally, for statistical function integration and estimation of response statistics associated with uniformly distributed random-variable inputs (uncorrelated), CVT has been found in initial investigations to provide superior points sets when compared against latin-hypercube and simple-random Monte Carlo methods and Halton and Hammersley quasi-random sequence methods. In this paper, the performance of all these sampling methods and a new variant (“Latinized” CVT) are further compared for non -uniform input distributions. Specifically, given uncorrelated normal inputs in a 2-D test problem, statistical sampling efficiencies are compared for resolving various statistics of response: mean, variance, and exceedence probabilities.

An optimization based domain decomposition method for partial differential equations

Abstract An optimization-based domain decomposition method for the solution of partial differential equations is presented. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. The existence of optimal solutions for the optimization problem is shown as is the convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as well as an eminently parallelizable gradient method for solving the optimality system. Then, the results of some numerical experiments and some concluding remarks are given. The latter includes the extension of the method to nonlinear problems such as the Navier-Stokes equations.

On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations

SummaryWe consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.

Insensitive Functionals, Inconsistent Gradients, Spurious Minima, and Regularized Functionals in Flow Optimization Problems

We use the simple context of Navier-Stokes flow in a channel with a bump to examine problems caused by the insensitivity of functionals with respect to design parameters, the inconsistency of functional gradient approximations, and the appearance of spurious minima in discretized functionals. We discuss how regularization can help overcome these problems. Along the way, we compare the discretize-then-differentiate and differentiate-then-discretize approaches to optimization, especially as they relate to the issue of inconsistent functional gradients. We close with a discussion of the implications that our observations have on more practical flow control and optimization problems.

Optimal Design and Control

This volume is a collection of papers written by engineers and mathematicians actively involved in innovative research in control and optimization, with emphasis placed on problems governed by partial differential equations. The papers arose from a workshop on control and optimization sponsored by SIAM. The volume presents research conducted at laboratory, industrial and academic institutions so that analyses, algorithms, implementations and applications are all well-presented in the papers. An overriding impression that ccan be gleaned from this volume is the complexity of problems addressed by not only those authors engaged in applications, but also by those engaged in algorithmic development and even mathematical analyses. Thus, in many instances, systematic approaches using fully nonlinear constraint equations are routinely used to solve control and optimization problems, in some cases replacing ad hoc or empirically based procedures. Many algorithmic issues are addressed, including sensitivity analyses, novel optimization methods especially tailored for specific applications, multilevel methods and programming techniques. Although many different applications (such as metal forging, heat transfer, contact problems, structures and acoustics) are considered in the volume, flow control plays a central role. Another recurring theme is shape control - that is, the use of shape of the boundary to effect control or to achieve optimal configuration. The book is intended for researchers and graduate students working in structural, fluid, thermal and electromagnetic design and control, or in the design and implementation of optimization algorithms.

On the Global Unique Solvability of Initial-Boundary Value Problems for the Coupled Modified Navier–Stokes and Maxwell Equations

Abstract.The global unique solvability of the first initial-boundary value problem in a bounded, two or three-dimensional domain with fixed perfectly conducting boundaries is proved for the modified Navier–Stokes equations coupled with the Maxwell equations. The system gives a deterministic description of the dynamics for conducting, incompressible, homogeneous fluids. Improved results are proved for the periodic boundary condition case.

On finite element approximations of problems having inhomogeneous essential boundary conditions

Abstract The analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered. The results are given for linear second order elliptic partial differential equations and for the nonlinear stationary Navier-Stokes equations. For certain easily implemented boundary treatments, optimal error estimates and numerical examples are provided for problems posed on polyhedral domains.