Abstract. Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments. 1. Introduction. Schwarz algorithms experienced a second youth over the last decades when distributed computers became more and more powerful and available. Fundamental convergence results for the classical Schwarz methods were derived for many partial differential equations and can now be found in several authoritative reviews [3,41,42] and books [34,33,39]. The Schwarz methods were also extended to systems of partial differential equations, such as the time harmonic Maxwell's equations [12,8], the time discretized Maxwell's equations [38], or to linear elasticity [18,19], but much less is known about the behavior of the Schwarz methods applied to hyperbolic systems of equations. This is true, in particular, for the Euler equations, to which the Schwarz algorithm was first applied in [31,32], where classical (characteristic) transmission conditions are used at the interfaces, or with more general transmission conditions in [7]. The analysis of such algorithms applied to systems proved to be very different from the scalar case; see [14,15].Over the last decade, a new class of overlapping Schwarz methods was developed for scalar partial differential equations, namely, the optimized Schwarz methods. These methods are based on a classical overlapping domain decomposition, but they use more effective transmission conditions than the classical Dirichlet conditions at the interfaces between subdomains. New transmission conditions were originally proposed for three different reasons: first, to obtain Schwarz algorithms that are convergent without...

Radiofrequency catheter ablation (RFCA) is a routine treatment for cardiac arrhythmias. During RFCA, the electrode-tissue interface temperature should be kept below 80°C to avoid thrombus formation. Open-irrigated electrodes facilitate power delivery while keeping low temperatures around the catheter. No computational model of an open-irrigated electrode in endocardial RFCA accounting for both the saline irrigation flow and the blood motion in the cardiac chamber has been proposed yet. We present the first computational model including both effects at once. The model has been validated against existing experimental results. Computational results showed that the surface lesion width and blood temperature are affected by both the electrode design and the irrigation flow rate. Smaller surface lesion widths and blood temperatures are obtained with higher irrigation flow rate, while the lesion depth is not affected by changing the irrigation flow rate. Larger lesions are obtained with increasing power and the electrode-tissue contact. Also, larger lesions are obtained when electrode is placed horizontally. Overall, the computational findings are in close agreement with previous experimental results providing an excellent tool for future catheter research.

In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.

Radiofrequency catheter ablation (RFCA) is an effective treatment for cardiac arrhythmias. Although generally safe, it is not completely exempt from the risk of complications. The great flexibility of computational models can be a major asset in optimizing interventional strategies, if they can produce sufficiently precise estimations of the generated lesion for a given ablation protocol. This requires an accurate description of the catheter tip and the cardiac tissue. In particular, the deformation of the tissue under the catheter pressure during the ablation is an important aspect that is overlooked in the existing literature, that resorts to a sharp insertion of the catheter into an undeformed geometry. As the lesion size depends on the power dissipated in the tissue, and the latter depends on the percentage of the electrode surface in contact with the tissue itself, the sharp insertion geometry has the tendency to overestimate the lesion obtained, especially when a larger force is applied to the catheter. In this paper we introduce a full 3D computational model that takes into account the tissue elasticity, and is able to capture the tissue deformation and realistic power dissipation in the tissue. Numerical results in FEniCS-HPC are provided to validate the model against experimental data, and to compare the lesions obtained with the new model and with the classical ones featuring a sharp electrode insertion in the tissue.

In the solution of Fluid-Structure Interaction problems, partitioned procedures are modular algorithms that involve separate fluid and structure solvers, that interact, in an iterative framework, through the exchange of suitable transmission conditions at the FS interface. In this work we study, using Fourier analysis, the convergence of partitioned algorithms based on Robin transmission conditions. We derive, for different models of the fluid and the structure, a frequency dependent reduction factor at each iteration of the partitioned algorithm, which is minimized by choosing optimal values of the coefficients in the Robin transmission conditions. Two-dimensional numerical results are also reported, which highlight the effectiveness of the optimization procedure.This allows one to introduce more performing Krylov methods for the solution of the FSI problem. In particular we mention the Dirichlet-Neumann/GMRES and the Robin-Neumann/GMRES schemes, which lead to different modular algorithms (see [2,4]).The introduction of Robin-Robin (RR) partitioned procedures in the framework of FSI problems, as generalization of the classic Dirichelt-Neumann (DN) scheme, has been motivated to overcome the limitations of the latter algorithm. In particular, the performances of the DN scheme when the added mass effect is high (that is when the fluid and structure densities are similar) are very poor, and a (sometimes big) relaxation is needed to reach convergence (see [6,12,23]). On the contrary, RR schemes highlighted better convergence properties in the presence of a high addedmass. In particular, in [2,3] the Robin-Neumann (RN) scheme has been shown to converge without relaxation, in the test cases studied, and to feature a big saving in computational time with respect to DN scheme.This behaviour has been mostly evidenced by numerical tests and only few theoretical results of the convergence properties of partitioned procedures for the FSI problems are available so far. In particular, at the best of the authors' knowledge, the only convergence analysis have been proposed in [6] for the DN scheme and in [3] for the RN scheme. We mention also the analysis in [4] and in [2] for DN-GMRES and RN-GMRES schemes. In all the cases, the analysis has been performed on a simplified problem where the fluid is described by a 2D potential flow and the structure by a 1D reduced model (the independent rings in [6] and the generalized string in the other works).The first goal of this work is to extend the convergence analysis of RR schemes (and then of DN) to more general classes of subproblems. In particular, we consider the generalized Stokes equations to describe the discretized-in-time fluid problem and a 2D linear elastic incompressible structure in the half plane. In particular, the proposed analysis are based on the application of the Fourier transform (see, e.g, [1,13,14]) and on the determination of a reduction factor.Secondly, we focus on parameters in Robin transmission conditions. Obviously, the convergence velocity of RR schemes he...