Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory

Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure

A-posteriori error estimates for linear exterior problems via mixed-FEM and DTN mappings

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions

Insect oral secretions suppress wound-induced responses in Arabidopsis

The induction of plant defences and their subsequent suppression by insects is thought to be an important factor in the evolutionary arms race between plants and herbivores. Although insect oral secretions (OS) contain elicitors that trigger plant immunity, little is known about the suppressors of plant defences. The Arabidopsis thaliana transcriptome was analysed in response to wounding and OS treatment. The expression of several wound-inducible genes was suppressed after the application of OS from two lepidopteran herbivores, Pieris brassicae and Spodoptera littoralis. This inhibition was correlated with enhanced S. littoralis larval growth, pointing to an effective role of insect OS in suppressing plant defences. Two genes, an ERF/AP2 transcription factor and a proteinase inhibitor, were then studied in more detail. OS-induced suppression lasted for at least 48 h, was independent of the jasmonate or salicylate pathways, and was not due to known elicitors. Interestingly, insect OS attenuated leaf water loss, suggesting that insects have evolved mechanisms to interfere with the induction of water-stress-related defences

FE–BE coupling for a transmission problem involving microstructure, in preparation

Abstract We analyze a finite element/boundary element procedure for a non-convex contact problem for the double-well potential. After relaxing the associated functional, the degenerate minimization problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which may then be solved numerically. The convergence of the Galerkin approximations to certain macroscopic quantities and a corresponding a posteriori estimate for the approximation error are discussed. Numerical results illustrate the performance of the proposed method

Domain Decomposition Methods For Boundary Integral Equations Of The First Kind: Numerical Results

: We present numerical experiments for the additive Schwarz algorithms applied to the h- and p-version Galerkin boundary element methods to solve the Laplacian and Helmholtz boundary value problems in two dimensions. Both weakly singular and hypersingular integral equations covering Dirichlet and Neumann problems, respectively, are considered. In the case of Laplacian problems where the Galerkin scheme yields symmetric and positive definite stiffness matrices, we use the preconditioned CG method, whereas in the case of Helmholtz problems we have indefinite non Hermitian stiffness matrices, and therefore we use the preconditioned GMRES method. We find that the two level additive Schwarz methods yield only logarithmically growing condition numbers for both the h- and p-versions; thus only a fixed number of iterations is necessary to compute appropriate approximations of the Galerkin solutions. We also perform multilevel methods for integral equations belonging to the boundary value probl..