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In this paper we develop an evolution of the C 1 virtual elements of minimal degree for the approximation of the Cahn-Hilliard equation. The proposed method has the advantage of being conforming in H 2 and making use of a very simple set of degrees of freedom, namely 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semi-discrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests. Introduction.The study of the evolution of transition interfaces, which is of paramount importance in many physical/biological phenomena and industrial processes, can be grouped into two macro classes, each one corresponding to a different method of dealing with the moving free-boundary: the sharp interface method and the phase-field method. In the sharp interface approach, the free boundary is to be determined together with the solution of suitable partial differential equations where proper jump relations have to imposed across the free boundary. In the phase field approach, the interface is specified as the level set of a smooth continuos function exhibiting large gradients across the interface.Phase field models, which date back to the works of Korteweg [33], Cahn and Hilliard [13,30,31], Landau and Ginzburg [34] and van der Waals [43], have been classicaly employed to describe phase separation in binary alloys. However, recently Cahn-Hilliard type equations have been extensively used in an impressive variety of applied problems, such as, among the others, tumor growth [47,39], origin of Saturn's rings [42], separation of di-block copolymers [15], population dynamics [17], image processing [9] and even clustering of mussels [35].Due to the wide spectrum of applications, the study of efficient numerical methods for the approximate solution of the Cahn-Hilliard equation has been the object of an intensive research activity. Summarizing the achievements in this field is a tremendous task that go beyond the scope of this paper. Here, we limit ourvselves to some remarks on finite element based methods, as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. As the Cahn-Hilliard equation is a fourth order nonlinear problem, a natural approach is the use of C 1 finite elements (FEM) as in [25,21]. However, in order to avoid the well known difficulty met in the implementation of C 1 finite elements, another possibility is the use of non-conforming (see, e.g., [22]) or discontinuous (see, e.g., [46]) methods; the drawback is that in such case the discrete solution will not satisfy a C 1 regularity. Alternatively, the most common strategy employed in practice to solve the Cahn-Hilliard equation with (continuos and discontinuous) finite elements is 2 to use mixed methods (see e.g. [23, 24] and [32] for the continuous and discontinuous setting, respecti...

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner for Isogeometric Analysis of scalar elliptic problems is constructed and analyzed by introducing appropriate discrete norms. A main result of this work is the proof that the proposed isogeometric BDDC preconditioner is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Another main result is the numerical validation of the theoretical convergence rate estimates by carrying out several two- and three-dimensional tests on serial and parallel computers. These numerical experiments also illustrate the preconditioner performance with respect to the polynomial degree and the regularity of the NURBS basis functions, as well as its robustness with respect to discontinuities of the coefficient of the elliptic problem across subdomain boundaries.

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The aim of this work is to investigate by means of numerical simulations the effects of myocardial deformation due to muscle contraction on the bioelectrical activity of the cardiac tissue. The three-dimensional electro-mechanical model considered consists of the following four components: the quasi-static orthotropic finite elasticity equations for the deformation of the cardiac tissue; the active tension model for the intracellular calcium dynamics and cross-bridge binding; the orthotropic Bidomain model for the electrical current flow through the tissue; the membrane model of the cardiac myocyte, including stretch-activated currents (I SAC ). In order to properly take into account cardiac mechanical feedbacks, the electrical current flow is described in a strongly coupled framework by the Bidomain model on the deformed tissue. We then derive a novel formulation of the Bidomain model in the reference configuration, with complete mechanical feedbacks affecting not only the conductivity tensors but also a convective term depending on the velocity of the deformation. The numerical simulations are based on our finite element parallel solver, which employs both Multilevel Additive Schwarz preconditioners for the solution of linear systems arising from the discretization of the Bidomain equations and Newton–Krylov-Algebraic Multigrid methods for the solution of nonlinear systems arising from the discretization of the finite elasticity equations. The results have shown that: (i) the I SAC current prolongs action potential duration (APD) of about 10–15 ms; (ii) the inclusion into the model of both I SAC current and the convective term reduces the dispersion of repolarization of about 7% (from 139 to 129 ms) and increases the dispersion of APD about three times (from 13 to 45 ms). These effects indicate that mechanical feedbacks might influence arrhythmogenic mechanisms when combined with pathological substrates.

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It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization\ud process towards local (and non-physical) minima, and by the accuracy of the method employed to discretize the underlying differential problem, which may not be able to\ud correctly capture the physics of the problem. In light of the above remarks, in this paper we consider polygonal meshes and employ the virtual element method (VEM) to solve\ud two classes of paradigmatic topology optimization problems, one governed by nearlyincompressible and compressible linear elasticity and the other by Stokes equations.\ud Several numerical results show the virtues of our polygonal VEM based approach with respect to more standard methods