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Non-overlapping Schwarz methods with generalized Robin transmission conditions were originally introduced by B. Després for time-harmonic wave propagation problems and have largely developed over the past thirty years. The aim of the paper is to provide both a review of the available formulations and methods as well as a consistent theory applicable to more general cases than studied until to date. An abstract variational framework is provided reformulating the original problem by the well-known form involving a scattering operator and an interface exchange operator, and the equivalence between the formulations is discussed thoroughly. The framework applies to a series of wave propagation problems throughout the de Rham complex, such as the scalar Helmholtz equation, Maxwell’s equations, a dual formulation of the Helmholtz equation in H(div), as well as any conforming finite element discretization thereof, and it applies also to coercive problems. Three convergence results are shown. The first one (using compactness) and the second one (based on absorption) generalize Després’ early findings and apply as well to the FETI-2LM formulation (a discrete method introduced by de La Bourdonnaye, Farhat, Macedo, Magoulés, and Roux). The third result, oriented on the work by Collino, Ghanemi, and Joly, establishes a convergence rate and covers cases with cross points, while not requiring any regularity of the solution. The key ingredient is a global interface exchange operator, proposed originally by X. Claeys and further developed by Claeys and Parolin, here worked out in full generality. The third type of convergence theory is applicable at the discrete level as well, where the exchange operator is allowed to be even local. The resulting scheme can be viewed as a generalization of the 2-Lagrange-multiplier method introduced by S. Loisel, and connections are drawn to another technique proposed by Gander and Santugini. [ABSTRACT FROM AUTHOR]

This article deals with the nonlinear eigenvalue problem originating from the finite element discretization of mechanical structures involving linear viscoelastic material. The material function is assumed to be positive real, which allows a location of the eigenvalues in the left complex half space of the Laplace domain. The solution method for the considered nonlinear eigenvalue problem is based on the contour integral method, where special focus is put on the efficient numerical computation of the linear system along the boundary of the given search area. For this purpose, the reduced order model technique is used and appropriate a priori error estimates are provided. Finally, the validity of the proposed method is illustrated in numerical examples. [ABSTRACT FROM AUTHOR]

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error $\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)}$ decreases like N, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results. [ABSTRACT FROM AUTHOR]

In this article, we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, all-floating FETI. We consider a scalar elliptic equation in a two- or three-dimensional domain with a highly heterogeneous (multiscale) diffusion coefficient. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. In other words, the subdomain partitioning does not need to resolve any jumps in the coefficient. Under suitable assumptions, we derive bounds for the condition numbers of one-level and all-floating FETI that are robust with respect to strong variations in the contrast in the coefficient, and that are explicit in some geometric parameters associated with the coefficient variation. In particular, robustness holds for face, edge, and vertex islands in high-contrast media. As a central tool we prove and use new weighted Poincaré and discrete Sobolev type inequalities that are explicit in the weight. Our theoretical findings are confirmed in a series of numerical experiments. [ABSTRACT FROM AUTHOR]