In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the “missing” boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero.

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).

We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.

A mixed finite element method for the Navier–Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier–Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf–sup conditions are developed.

The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrödinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in [Schneider, 2009]. Recently, we globalized the CC formulation to the full continuous space, giving a root equation for an infinite dimensional, nonlinear Coupled Cluster operator that is equivalent the full electronic Schrödinger equation [Rohwedder, 2011]. In this paper, we combine both approaches to prove existence and uniqueness results, quasi-optimality estimates and energy estimates for the CC method with respect to the solution of the full, original Schrödinger equation. The main property used is a local strong monotonicity result for the Coupled Cluster function, and we give two characterizations for situations in which this property holds.

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.

One of the current debate about simulating the electrical activity in the heart is the following: Using a realistic anatomical setting, i.e. realistic geometries, fibres orientations, etc., is it enough to use a simplified 2-variable phenomenological model to reproduce the main characteristics of the cardiac action potential propagation, and in what sense is it sufficient? Using a combination of dimensional and asymptotic analysis, together with the well-known Mitchell − Schaeffer model, it is shown that it is possible to accurately control (at least locally) the solution while spatial propagation is involved. In particular, we reduce the set of parameters by writing the bidomain model in a new nondimensional form. The parameters of the bidomain model with Mitchell − Schaeffer ion kinetics are then set and shown to be in one-to-one relation with the main characteristics of the four phases of a propagated action potential. Explicit relations are derived using a combination of asymptotic methods and ansatz. These relations are tested against numerical results. We illustrate how these relations can be used to recover the time/space scales and speed of the action potential in various regions of the heart.

We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.