Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

Stable boundary element domain decomposition methods for the Helmholtz equation

In this paper, we present a stable boundary element domain decomposition method to solve boundary value problems of the Helmholtz equation via a tearing and interconnecting approach. A possible non-uniqueness of the solution of local boundary value problems due to the appearance of local eigensolutions is resolved by using modified interface conditions of Robin type, which results in a Galerkin boundary element discretization which is robust for all local wave numbers. Numerical examples confirm the stability of the proposed approach.     

Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

In this paper, we consider the inverse Robin transmission problem with one electrostatic measurement. We prove a uniqueness result for the simultaneous determination of the Robin parameter p, the conductivity k, and the subdomain D, when D is a ball. When D and k are fixed, we prove a uniqueness result and a directional Lipschitz stability estimate for the Robin parameter p. When p and k are fixed, we give an upper bound to the subdomain D. For the reconstruction purposes of the Robin parameter p, we set the inverse problem under an optimization form for a Kohn–Vogelius cost functional. We prove the existence and the stability of the optimization problem. Finally, we show some numerical experiments that agree with the theoretical considerations. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical estimation of the piecewise constant Robin coefficient by identifying its discontinuous points

We propose a new numerical method for estimating the piecewise constant Robin coefficient in two-dimensional elliptic equation from boundary measurements. The Robin inverse problem is recast into a minimization of an output least-square formulation. A technique based on determining the discontinuous points of the unknown coefficient is suggested, and we investigate the differentiability of the solution and the objective functional with respect to the discontinuous points. Then we apply the Gauss-Newton method for reconstructing the shape of the unknown Robin coefficient. Numerical examples illustrate its efficiency and stability.     

Boundary element solution of a scattering problem involving a generalized impedance boundary condition

A boundary element method is introduced to approximate the solution of a scattering problem for the Helmholtz equation with a generalized Fourier–Robin‐type boundary condition given by a second‐order elliptic differential operator. The formulation involves three unknown fields, but is free from any hypersingular integral. Existence and uniqueness of the solution are established using a Babuška inf–sup condition. When implementing the method, a lumping process allows to remove two fields from the formulation. The numerical solution has thus the same cost as the one of a problem relative to a usual Neumann boundary condition. Numerical tests confirm the ability of the method for solving this type of non‐standard boundary value problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Unconditionally stable modified methods for the solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions

In this article, we discuss modified three level implicit difference methods of order two in time and four in space for the numerical solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions. Ghost points are introduced to obtain fourth‐order approximations for boundary conditions. Matrix stability analysis is carried out to prove stability of the method for telegraphic equations in two and three dimensions with Neumann boundary conditions. Numerical experiments are carried out and the results are found to be better when compared with the results obtained by other existing methods.     

On output functionals of boundary value problems on stochastic domains

We consider the computation of output functionals of random solutions to elliptic boundary value problems in domains with random boundary perturbations. We use a second‐order shape calculus to linearize the problem around a fixed nominal domain. For known mean and two‐point correlation function of the boundary perturbation, we derive, with leading order, deterministic expressions for the mean and the variance of the random output functional. These expressions include the solution of the boundary value problem on the nominal domain and a further, deterministic solution of the so‐called adjoint equation. The theoretical findings are supported and quantified by numerical experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

Coercivity for elliptic operators and positivity of solutions on Lipschitz domains

We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from W ^−1,2 are identified as positive measures.     

Reduced multiscale computation on adapted grid for the convection-diffusion Robin problem

We propose a reduced multiscale finite element method for a convection-diffusion problem with a Robin boundary condition. The small perturbed parameter would cause boundary layer oscillations, so we apply several adapted grids to recover this defect. For a Robin boundary relating to derivatives, special interpolating strategies are presented for effective approximation in the FEM and MsFEM schemes, respectively. In the multiscale computation, the multiscale basis functions can capture the local boundary layer oscillation, and with the help of the reduced mapping matrix we may acquire better accuracy and stability with a less computational cost. Numerical experiments are provided to show the convergence and efficiency.     

Multiple solutions of semilinear degenerate elliptic boundary value problems

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super‐sub‐solution method in the degenerate case, and is distinguished by the extensive use of an L^p Schauder theory elaborated for second‐order, elliptic differential operators with discontinuous zero‐th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub‐ and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the Optimal Insulation of Conductors

We coat a conductor with an insulator and equate the effectiveness of this procedure with the rate at which the body dissipates heat when immersed in an ice bath. In the limit, as the thickness and conductivity of the insulator approach zero, the dissipation rate approaches the first eigenvalue of a Robin problem with a coefficient determined by the shape of the insulator. Fixing the mean of the shape function, we search for the shape with the least associated Robin eigenvalue. We offer exact solutions for balls; for general domains, we establish existence and necessary conditions and report on the results of a numerical method.     

Generalized exterior boundary-value problems and optimization for the Helmholtz equation

A class of radiation problems is considered for the Helmholtz equation in exterior domains bounded by a smooth surface on which Dirichlet, Neumann, or Robin boundary conditions are imposed. The problem of finding the boundary data which maximizes far field power in a restricted subset of far field directions is formulated as a constrained maximization problem. Existence of an optimal solution in a variety of control domains is established. The particular case when the boundary is circular and the control domain is the unit ball inL ² is treated in detail. An algorithm for constructing the optimal solution is derived and used to obtain explicit numerical results.This work was supported by the US Air Force under Grant No. AFOSR 81-0156. The work was completed while the first author was on leave to the Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, BRD.     

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound‐soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.     

Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case

In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupling of FEM and BEM in Shape Optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L ²-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation     

Shape analysis in membrane vibration

In order to characterize the domain Ω minimizing the normal stress on the boundary of a membrane, we are concerned with the shape derivative of the functional \def\d{\,{\rm d}}$J(\Omega)=\int_I\int_{\partial\Omega}(\partial y/\partial n)^2\,g\d x\d t$ , where I is the time interval, y is the solution to the wave equation and g a weight coefficient. We first recall some results on the transformation of domains and investigate the shape derivative of the state. Then we compute the derivative of J with respect to the domain. Eventually, we give a necessary condition of optimality which relies heavily on the oriented distance function and its properties around the neighbourhood of the boundary. Copyright © 2000 John Wiley & Sons, Ltd.     

Robin spectral rigidity of nearly circular domains with a reflectional symmetry

This is a note on a paper of De Simoi–Kaloshin–Wei. We show that by combining their techniques with the wave trace invariants of Guillemin–Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of De Simoi–Kaloshin–Wei to the case of Robin boundary conditions. We will consider the same generic subset as did by De Simoi–Kaloshin–Wei of smooth strictly convex ?₂-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary ?₂-symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.     

Maximax and minimax rearrangement optimization problems

In this paper we present a general theory concerning two rearrangement optimization problems; one of maximization and the other of minimization type. The structure of the cost functional allows to formulate the two problems as maximax and minimax optimization problems. The latter proves to be far more interesting than the former. As an application of the theory we investigate a shape optimization problem which has already been addressed by other authors; however, here we prove our method is more efficient, and has the advantage that it captures more features of the optimal solutions than those obtained by others. The paper ends with a special case of the minimax problem, where we are able to obtain a minimum size estimate related to the optimal solution.