# Abstracts

### < Session Nitsche / XFEM | Session A posteriori | Session Artificial BC / DDM >

• Session Nitsche / eXtended finite elements (organiser : Franz Chouly)

Roland Becker, Université de Pau
Nitsche’s method for incompressible flow

Stéphane Bordas, Université du Luxembourg
Extended Finite Element Method with Global Enrichment (with K. Agathos, E. Chatzi, D. Talaslidis)

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The major problem associated with geometrical enrichment is that it significantly deteriorates the conditioning of the resulting system matrices, thus increasing solution times and in some cases making the systems unsolvable. For 2D problems this can be dealt with by employing degree of freedom gathering [1] which essentially inhibits spatial variation of enrichment function weights. However, for the general 3D problem such an approach is not possible since spatial variation of the enrichment function weights in the direction of the crack front is necessary in order to reproduce the variation of solution variables, such as the stress intensity factors, along the crack front. The proposed method solves the above problem by employing a superimposed mesh of special elements which serve as a means to provide variation of the enrichment function weights along the crack front while still not allowing variation in any other direction. The method is combined with special element partitioning algorithms [2] and numerical integration schemes [3] as well as techniques for the elimination of blending errors between the standard and enriched part of the approximation in order to further improve the accuracy of the produced results.

Additionally, a novel benchmark problem is introduced which enables the computation of displacement and energy error norms as well as errors in the stress intensity factors for the general 3D case. Through this benchmark problem it is shown that the proposed method provides optimal convergence rates, improved accuracy and reduced computational cost compared to standard XFEM.

Erik Burman, University College London
Nitsche’s method for multiphysics problems on unfitted meshes (with Thomas Boiveau, Susanne Claus, Miguel Fernández, Peter Hansbo, Mats Larson, Andre Massing, Luke Swift)

In this talk we will discuss recent advances on the use of Nitsche’s method as a tool for multiphysics coupling in computational mechanics. First we will briefly review the basic concepts of Nitsche’s method with special focus on unfitted or fictitious domain methods. Some different applications will be discussed such as coupling of different models of elasticity and coupling of Helmholtz equations over nonmatching meshes. Finally we will consider some different formulations for fluid strucuture interaction, that have in common the possibility of designing loosely coupled schemes for fully partitioned time marching. Numerical experiments indicate that a nonsymmetric version of Nitsche’s method, without penalty term, has some special properties in this framework and we will present some of our recent results on this topic.

Keywords : unfitted finite element methods, fictitious domain, Nitsche’s method, fluid-structure interaction

Alfonso Caiazzo, WIAS Berlin
Modeling and simulation of fluid flows through a porous interface

This talk focuses on a numerical method to simulate an incompressible fluid through an immersed porous interface. The interface is modeled via conformal surface in the computational mesh, taken into account by a surface measure term in the Navier–Stokes equations, depending on a resistance parameter. This approach can be used, for example, to model valves or to simulate blood flood through an immersed stent. First, the stability analysis for a monolithic formulation of the Stokes equation is discussed, showing that the standard Pressure Stabilized Petrov–Galerkin (PSPG) finite element method is stable, and optimally convergent. As next, a fractional step algorithm is derived. For this case, we show that an appropriate Nitsche treatment of the interface condition allows for uniform energy stability in time for any nonnegative value of the interface resistance. Finally, the proposed projection scheme is employed for the simulation of blood flows through immersed stents and for the modeling of the pulmonary valve.

Daniela Capatina, LMAP, Université de Pau
Nitsche’s Extended Finite Element Method for a Fracture Model in Porous Media

We study the numerical approximation of Darcy equations in a fractured domain, with highly discontinuous permeability tensors. In order to avoid the meshing of the fracture, we work with the asymptotic model obtained as the fracture’s thickness goes to 0 in the weak formulation. This leads to treat a non-standard transmission condition between the two domains delimited by the fracture, instead of solving the equations in the fracture. Then we propose an approximation of the asymptotic model based on Nitsche’s extended finite element method, which allows to consider fractures not aligned with the mesh. We prove consistency and stability of the discrete formulation and we present some numerical tests showing the relevance of the method. Finally, we discuss an extension of this work to other model problems. This is a joint work with R. luce and H. El-Otmany.

Miguel Fernández, INRIA Paris-Rocquencourt
Unfitted mesh methods and coupling schemes for incompressible fluid-structure interaction

Fictitious domain/immersed boundary methods for the numerical simulation of fluid-structure interaction problems involving large interface deflections have recently seen a surge of interest. Most of the existing approaches are known to be inaccurate in space either because the fluid equations are integrated in a non-physical (fictitious) domain or because the discrete approximations are not able to reproduce weak and strong discontinuities of the physical solution. In this talk, we present alternative unfitted formulations which circumvent these accuracy issues. The kinematic/kinetic fluid-solid coupling is enforced consistently using a variant of Nitsche’s method involving cut elements. Robustness with respect to arbitrary interface/element intersections is guaranteed through suitable stabilization. Whenever present, weak and strong discontinuities across the interface are allowed via suitable XFEM enrichment. Several coupling schemes, with different degrees of fluid-solid splitting (implicit, semi-implicit and explicit), will be discussed. A series of numerical tests, involving static and moving interfaces, illustrates the performance of the different methods proposed.

Patrick Hild, Université Toulouse 3
Adaptation de la méthode de Nitsche aux problèmes de contact
avec ou sans frottement en élastodynamique
(avec F. Chouly et Y. Renard)

Dans cette présentation, on s’intéresse au problème de contact dynamique en élasticité avec ou sans frottement. On propose une adaptation de la méthode de Nitsche pour la semi-discrétisation en espace par éléments finis du problème. Les résultats présentés concernent l’existence et l’unicité des problèmes semi- et totalement discrétisés, la stabilité des schémas et les simulations correspondantes.

Joaquin Mura, Pontificia Universidad Católica de Valparaíso
Numerical simulation of the fluid-structure interaction in stended aneurysms (with M. A. Fernández & J.-F. Gerbeau)

An aneurysm is a condition whereby the weakened artery wall may dilates to dangerous proportions. One of the less invasive treatments is to insert a stent, which is a medical device introduced into the artery lumen in order to diminish the pressure on the aneurysm wall and so avoiding its rupture.
In this article we perform the simulation of the fluid-structure interaction between the blood, the artery wall and the stent. We consider the blood modeled by incompressible Navier-Stokes equations in ALE formulation, while the solid parts are modeled using nonlinear shells in a Lagrangian framework. We suppose the stent being perfectly matched with the artery, i.e., with no filtration or endoleaks, such that we obtain two disconnected regions of fluid by the stent wall. The main difficulty in this problem is the numerical solution for the isolated portion of fluid, where the explicit coupling with standard Dirichlet-Neumann boundary conditions may lead to non uniqueness for the intra-aneurysmal pressure and also to violate the divergence free condition. To solve this inconvenient we use Robin-Neumann conditions for the coupling, which allow us not only to obtain successful numerical simulations but also a less sensitive scheme to the added-mass effect and a more stable and robust iterative technique than the Dirichlet-Neumann procedure (see e.g. [1],[2],[3]).
We present numerical evidence of the well-posedeness of this technique and several examples showing the effect of the stent on the aneurysm wall.

Yves Renard, INSA Lyon
Approximation des conditions de contact en élastostatique :
méthode de Nitsche et lien avec le lagrangien augmenté proximal

L’objectif de cette présentation est de faire le point sur les différentes stratégies de prise en compte consistante des conditions de contact/frottement entre structures élastiques. On fera notamment la comparaison entre la méthode de Nitsche, qui a été étendue récemment aux conditions de contact, et la méthode du lagrangien augmenté proximal. On discutera de la genèse théorique des différentes méthodes, de leurs éventuels liens et on essaiera de comparer leur intérêt pratique dans le
cadre de la discrétisation par éléments finis et la résolution numérique dans le cadres statique.

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• Session A posteriori estimates in space and time (organiser : Alexei Lozinski)

Christine Bernardi, Laboratoire Jacques-Louis Lions, Université Paris 6
Richards model for unsaturated porous media

We consider the equation due to Richards which models the water flow in a partially saturated underground porous medium under the surface, with mixed boundary conditions involving Signorini conditions when it rains too much. We propose a discretization of this equation by an implicit Euler’s scheme in time and mixed finite elements in space. We perform the a posteriori analysis of this discretization, in order to improve its efficiency via time step and mesh adaptivity. Some numerical experiments confirm the interest of this approach.

Andreas Brenner, Applied Mathematics III, University Erlangen—Nuernberg
A-posteriori estimates for the rotational pressure-correction projection method

We present a-posteriori error estimates for the time discretization of the incompressible instationary Stokes equation by the two-step backward differential formula method (BDF2) for the rotational pressure-correction projection method.
As introduction we explain our techniques for the fully-discrete Stokes equations and give an short overview of projection methods.

This is a joint work with Eberhard Bänsch.

Olga Gorynina, Laboratoire des Mathématiques de Besançon, Université de Franche-Comté
A posteriori error estimates for the wave equation discretized in time with the Newmark scheme

We develop a posteriori error estimates in space and time for the wave equation discretized by the Newmark scheme (of second order in time) and by finite elements in space. We start by reformulating the wave equation as a first-order system, as in Bernardi & Süli M3AS 2005. We observe then that the Newmark scheme can be reinterpreted as the Crank-Nicolson discretization of the reformulated system so that the techniques from Lozinski Picasso Prachittham SISC 2009 (using a piecewise quadratic polynomial reconstruction of the numerical solution) can be applied. Optimal a posteriori error estimates in time can be thus recovered, and the estimates in space easily follow. We shall present the technical proofs and illustrate them by some preliminary numerical results. This is a joint work with A. Lozinski and M. Picasso.

Foteini Karakatsani, Department of Mathematics, University of Chester
A posteriori error estimates for fully discrete fractional-step $\theta$-approximations for parabolic equations

We derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretization in time we apply the fractional-step $\theta$-scheme and for the discretization in space the finite element method with finite element spaces that are allowed to change with time. The first optimal order a posteriori error estimates for the norms of $L^\infty(0,T ;L^2(\varOmega))$ and $L^2(0,T ;H^1(\varOmega))$ are derived by applying the reconstruction technique.

Irene Kyza, Division of Mathematics, University of Dundee
On the design and a posteriori error control for higher order in time ALE formulations

Arbitrary Lagrangian Eulerian (ALE) formulations are useful when dealing with problems defined on deformable domains, such as fluid-structure interactions. Having at hand higher order (at least second order) in time ALE methods is important, as they lead to realistic simulations involving fluids in 3d. However such methods are very limited in the literature.

In this talk we present some recent results on the design, the stability and the a posteriori error control of discontinuous Galerkin ALE methods of any order, for an evolution convection-diffusion model problem on time-dependent domains. Exploiting the variational structure of the dG method we prove that our dG schemes enjoy the same stability properties as the continuous problem. The same is true for the practical Reynolds’ methods which result from the dG methods after numerical integration by quadratures which inherit a discrete Reynolds’ identity. We also study the stability properties of Runge-Kutta-Radau methods ; these methods are proven to be stable under a mild constraint on the time-step related to the motion of the domain.

Finally, using an appropriate extension of the reconstruction technique on time-dependent domains and the ALE framework, we provide a posteriori error estimates for the proposed methods.
Key ingredients for the analysis are the definition of an appropriate reconstruction and the extension of certain projections on time-dependent domains. The proposed reconstruction is a generalisation of the dG time reconstruction, introduced earlier by Makridakis & Nochetto for the corresponding equations on time independent domains. Using PDE techniques, as for the original problem written in the ALE framework, we manage to prove optimal order a posteriori error bounds. The a posteriori error control gives important information on the behaviour of the error with respect to the movement of the domain. In particular, our analysis allows variable time steps and suggests that time adaptivity is essential for highly oscillatory ALE maps. Numerical experiments illustrate our theoretical results.

This is joint work with A. Bonito from Texas A&M University (USA) and R.H. Nochetto from the University of Maryland (USA).

Omar Lakkis, Department of Mathematics, University of Sussex
Aposteriori error analysis of timestepping schemes for the wave equation

Aposteriori error estimates provide a rigorous foundation for the derivation of efficient adaptive algorithms for the approximation of solutions of partial differential equations. While the literature abounds with results for elliptic and (more recently) parabolic equations, the situation is much less developed for the hyperbolic equations such as the wave equation. In this talk, I will review some of the "standard" aposteriori results by Bangerth, Rannacher, Bernardi, and Süli, for the wave equation and present recent developments and improvements. Particular focus will be given to practically relevant methods such as Verlet, or Cosine, methods, a popular example of which is the Leap-frog method.

This is based on joint work with E.H. Georgoulis, C. Makridakis and J.M. Virtanen.

Marco Picasso, MATHICSE, EPFL
A sharp error estimator for the transport equation with anisotropic, stabilized finite elements and the Crank-Nicolson scheme (avec Samuel Dubuis)

The transport equation in two space dimensions is considered. Stabilized finite elements are used for the space discretization, as in Burman CMAME 2010, the Crank Nicolson scheme is used for the time discretization. The mesh triangles may have large aspect ratio whenever needed.

Following Lozinsky Picasso Prachittham SISC 2009, a space/time error estimator is derived, order two in time. An adaptive space/time algorithm is presented and numerical results confirm the sharpness of the error estimator.

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• Session Artificial Boundary Conditions / Domain Decomposition (organiser : Pauline Klein)

Xavier Antoine, IECL, Université de Lorraine
Quasi-Optimal Domain Decomposition Methods for Harmonic Waves

The aim of this talk is to present and compare optimized Schwarz domain decomposition methods with various transmission conditions for solving the time-harmonic acoustic (Helmholtz) and electromagnetic (Maxwell) equations. We will analyze how to build suitable transmission conditions which are particularly well-adapted to the high frequency regime. Various 2D and 3D simulations will be presented. These examples are based on the use of the freely available DDM solver, called GetDDM (http://onelab.info/wiki/GetDDM), developed between the Institut Elie
Cartan de Lorraine and the Institut Montéfiore, Liège.

Christophe Besse, MIT, Université de Toulouse
Artificial boundary conditions for dispersive equations

We will propose in this talk an introduction to the derivation and the use of artificial boundary conditions for dispersive equation. We will insist in the boundary conditions for the Schrödinger and the linearized Korteweg de Vries equations.

Martin J. Gander, Section de Mathématiques, Université de Genève
Sweeping Preconditioning, Source Transfer and optimized Schwarz Methods

Absorbing boundary conditions and perfectly matched layers are not only useful for the truncation of computational domains, they can also be very effectively used to obtain preconditioners. This was first realized in the context of Schwarz methods about 20 years ago, in the research group around Frederic Nataf and Laurence Halpern, and led to the class of optimized Schwarz methods. Around the same time, very similar mathematical concepts also appeared in approximate factorizations, which led to the class of frequency filtering and AILU preconditioners. More recent interest in these methods was sparked by the difficulty to solve Helmholtz and Maxwell problems by iterative methods, and the introduction of the sweeping preconditioners by Engquist et al. and source transfer domain decomposition methods by Chen et al. I will present the relation between all these techniques in the context of optimal and optimized Schwarz methods.

Véronique Martin, LAMFA, Université de Picardie Jules Verne
A new algorithm for heterogeneous domain decomposition

Often computational models are too expensive to be solved in the entire domain of simulation, and a cheaper model would suffice away from the main zone of interest. We present for the concrete example of an evolution problem of advection reaction diffusion type a heterogeneous domain decomposition algorithm which allows us to recover a solution that is very close to the solution of the fully viscous problem, but solves only an inviscid problem in parts of the domain. Our new algorithm is based on specific absorbing boundary conditions.

In this talk, I will first give a review of existing heterogeneous domain decomposition algorithms. Then I will present our new algorithm and I will give an error analysis. Numerical results will also be shown.

This is a joint work with L. Halpern and M. Gander.

Antoine Tonnoir, POEMS, ENSTA-ParisTech
A domain decomposition approach for solving scattering problems in unbounded anisotropic media

The general motivation of this work is the numerical simulation of Non Destructif Testing using ultrasonic waves. The goal is to design a method to compute the diffraction of elastic waves in time-harmonic regime by a bounded defect. The difficulty is to take into account the infinite domain as well as reducing the finite element (FE) calculations to a bounded area around the defect. This is a difficult problem due to the anisotropy and, in particular, classical methods such as Perfectly matched Layers fails.

In this talk, we consider the diffraction problem in a 2D anisotropic medium infinite in the two directions. The key point is that we can compute (via the Fourier Transform) the solution in a half-plane knowing its trace on the boundary. Then, our idea is to combine the FE representation arround the defect with several representations of the solution in several half-planes that surround the defect. This leads us to a formulation coupling, via integral operators, both the solution in a bounded domain and the traces on the boundaries of the half-planes.

The method has been implemented an tested using a C++ code, first for a simpler scalar equation, and then extended to the case of the vectorial equation of elasticity. Numerical results will be shown.

This is a joint work with A.-S. Bonnet-Ben Dhia and S. Fliss.

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