Steklov-Poincare Operator Research Articles

Steklov–Poincaré Operator for A System of Coupled Abstract Cauchy Problems

We are interested in a coupled system of two abstract Cauchy problems; one of them is posed in small time interval. The aim is to obtain a condition that replaces the effect of this small interval. This condition is constructed from an approximation of Steklov–Poincare operator. In this paper, we analyze the construction and the approximation of this operator. Finally, the results obtained will be applied to a problem of diffraction of an electromagnetic wave by a perfectly conducting planar obstacle coated with thin layer of dielectric material.

Darboux Moutard Transformations and Poincaré—Steklov Operators

Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.

Finite-Dimensional Approximations of the Steklov–Poincare Operator in Periodic Elastic Waveguides

Finite-Dimensional Approximations of the Steklov–Poincare Operator in Periodic Elastic Waveguides

Finite-Dimensional Approximations of the Steklov–Poincaré Operator in Periodic Elastic Waveguides

For anisotropic elastic waveguides with cylindrical or periodic outlets to infinity, artificial integro-differential conditions are developed at the end face of a truncated waveguide, which simulate the Steklov–Poincare operator for scalar problems. Asymptotically sharp error estimates are derived in the definition of both the elastic fields themselves in the waveguide and the corresponding scattering coefficients.

Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides

We consider the Dirichlet and Neumann problems for the Laplace operator in periodic waveguides. Integro-differential connections between the solution and its normal derivative, interpreted as a finite-dimensional version of the Steklov–Poincare operator, are imposed on the artificial face of the truncated waveguide. These connections are obtained from the orthogonality and normalization conditions for the Floquet waves which are oscillating incoming/outgoing, as well as exponentially decaying/growing in the periodic waveguide. Under certain conditions, we establish the unique solvability of the problem and obtain error estimates for the solution itself, as well as for scattering coefficients in the solution. We give examples of trapped waves in periodic waveguides.

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by <i>d</i>-sets

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincare-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for $n$ and $d$-sets.

Numerical Analysis of the Stress-Strain State of a Body with Thin Inclusion by the Domain Decomposition Method

We consider a mathematical model of elastic body with thin inclusion or coating in the form of a thin elastic shell. It is shown that the corresponding Steklov–Poincare operator of the mathematical model possesses the properties guaranteeing the existence and uniqueness of a weak solution of the boundary-value problem. We propose a method of solution based on the domain decomposition algorithm with the use of the boundary-element and finite-element methods. We prove the convergence of the iterative domain decomposition method and present the results of numerical experiments.

One-sided contact problems with friction arising along the normal

We study a boundary contact problem for a micropolar homogeneous elastic hemitropic medium with regard of friction; in the considered case, friction forces do not arise in the tangential displacement but correspond to a normal displacement of the medium. We consider two cases: the coercive case (in which the elastic body has a fixed part of the boundary) and the noncoercive case (without fixed parts). By using the Steklov–Poincare operator, we reduce this problem to an equivalent boundary variational inequality. Existence and uniqueness theorems are proved for the weak solution on the basis of properties of general variational inequalities. In the coercive case, the problem is unconditionally solvable, and the solution depends continuously on the data of the original problem. In the noncoercive case, we present closed-form necessary conditions for the existence of a solution of the contact problem. Under additional assumptions, these conditions are also sufficient for the existence of a solution.

Boundary augmented Lagrangian method for the Signorini problem

An augmented Lagrangian method, based on boundary variational formulations and fixed point method, is designed and analyzed for the Signorini problem of the Laplacian. Using the equivalence between Signorini boundary conditions and a fixed-point problem, we develop a new iterative algorithm that formulates the Signorini problem as a sequence of corresponding variational equations with the Steklov-Poincare operator. Both theoretical results and numerical experiments show that the method presented is efficient.

The equivalent circuit of a joining two-port

An equivalent circuit of a joining two-port is proposed. The two-port provides for the stability and convergence of the results of semi-natural modeling of a partitioned linear dynamical system to the results of numerical modeling of the integral original system. The functional properties of the obtained two-port are investigated. It is shown that the obtained results are coupled with the well-known results in the decomposition methods and system modeling by parts. It is supposed that the obtained two-port is one of the simplest hypostases of the Poincare–Steklov operator.

Space-Time Domain Decomposition for Reduced Fracture Models in Mixed Formulation

In this paper we are interested in the fast path fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincare operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.

Contact Pressures and Cracks Identification by using the Dirichlet-to-Neumann Solver in Elasticity

In this work we present a numerical data completion method based on the Dirichlet-to-Neumann algorithm, by working in a linear elasticity framework. We begin by recasting the problem in terms of the Steklov-Poincare operator which is commonly used in domain decomposition. Then we present the Dirichlet-to-Neumann algorithm and state the equivalence between both formulations. The proposed method is applied to identify a contact pressures distribution and interfacial cracks.

The Poincaré–Steklov Operator in Hybrid Finite Element-Boundary Integral Equation Formulations

The Poincare-Steklov operator provides a direct relation between the tangential electric and magnetic field at the boundary of a simply connected domain, and a discrete equivalent of the operator can be constructed from the sparse finite element (FE) matrix of that domain by forming the Schur complement to eliminate the interior unknowns. Identifying the FE system matrix as a discretized version of the Poincare-Steklov operator allows us to describe and analyze FE and hybrid finite element-boundary integral equation (FE-BIE) formulations from an operator point of view. We show how this operator notation provides substantial theoretical insight into the analysis of spurious solutions in hybrid FE-BIE methods, and we apply the theory on a TM scattering example to predict the breakdown frequencies of different hybrid formulations.

Construction of direct and iterative decomposition methods

In order to solve boundary value problems with the Dirichlet-Dirichlet type conditions by decomposing the computational domain into subdomains joined without overlapping, the discrete Green’s functions are used to approximate the Poincare-Steklov operator equation on the interface of the subdomains, which involves the difference of the normal derivatives of the solution on the opposite sides of the interface. Basing on this approach, some direct and iterative decomposition methods are constructed that are basically of a parallel nature. Examples of numerical computations demonstrate the accuracy and convergence of the algorithms.

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare–Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1 − O ( h 1 / 4 ) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.

Numerical analysis of the Navier–Stokes/Darcy coupling

We consider a differential system based on the coupling of the Navier–Stokes and Darcy equations for modeling the interaction between surface and porous-media flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov–Poincare operators, and we prove its well-posedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincare–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump.

Sensivity Analysis of Infinite Order Parabolic Optimal Control Problems

Abstract In the paper the first order sensitivity analysis is performed for a class of optimal control problems for infinite order parabolic equations. A singular perturbation of geometrical domain of integration is introduced in the form of a circular hole. The Steklov-Poincare operator on a circle is defined in order to reduce the problem to regular perturbations in the truncated domain. The optimality system is differentiated with respect to the small parameter and the directional derivative of the optimal control is obtained as a solution to an auxiliary optimal control problem.

Local boundary estimates for the Lagrange multiplier discretization of a Dirichlet boundary value problem with application to domain decomposition

We give an estimate on the error resulting from approximating the outer normal derivative of the solution of a second-order partial differential equation with the Lagrange multiplier obtained in using the Lagrange multiplier method for imposing the Dirichlet boundary conditions. We consider both the case of smooth domains and, in view of an application in the framework of domain decomposition, the case of polygonal domains. The estimate given, supported by numerical results, shows that the mesh in the interior of the (sub)domain can be chosen more coarsely than near the boundary, when only a good approximation of the outer normal derivative is needed, as in the case of the evaluation of the Steklov-Poincare operator, or when solving with a Schur complement approach the linear system arising from the three-fields domain decomposition method. Keywords: Lagrange multipliers, error estimate, normal derivative, Steklov-Poincare operator

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincare–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.