Category: News

Lars Diening: 23 February at 10 am, A415 Inria Paris. This is a joint work with Massimo Fornasier and Maximilian Wank. In this talk we propose a iterative method to solve the non-linear -Poisson equation. The method is derived from a relaxed energy by an alternating direction method. We are able to show algebraic convergence of the iterates to the solution. However, our numerical experiments based on finite elements indicate optimal, exponential convergence.

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The Multiscale Finite Element Method (MsFEM) is a powerful numerical method in the context of multiscale analysis. It uses basis functions which encode details of the fine scale description, and performs in a two-stage procedure: (i) offline stage in which basis functions are computed solving local fine scale problems; (ii) online stage in which a cheap Galerkin approximation problem is solved using a coarse mesh. However, as in other numerical methods, a crucial issue is to certify that a prescribed accuracy is obtained for the numerical solution. In the present work, we propose an a posteriori error estimate for MsFEM using the concept of Constitutive Relation Error (CRE) based on dual analysis. It enables to effectively address global or goal-oriented error estimation, to assess the various error sources, and to drive robust adaptive algorithms. We also investigate the additional use of model reduction inside the MsFEM strategy in order to further decrease numerical costs. We particularly focus on the use of the Proper Generalized Decomposition (PGD) for the computation of multiscale basis functions. PGD is a suitable tool that enables to explicitly take into account variations in geometry, material coefficients, or boundary conditions. In many configurations, it can thus be efficiently employed to solve with low computing cost the various local fine-scale problems associated with MsFEM. In addition to showing performances of the coupling between PGD and MsFEM, we introduce dedicated estimates on PGD model reduction error, and use these to certify the quality of the overall MsFEM solution.

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Amina Benaceur: January 26 at 3pm, A415 Inria Paris. We address the reduced order modeling of parameterized transient non-linear and non-affine partial differential equations (PDEs). In practice, both the treatment of non-affine terms and non-linearities result in an empirical interpolation method (EIM) that may not be affordable although it is performed `offline’, since it requires to compute various nonlinear trajectories using the full order model. An alternative to the EIM that lessens its cost for steady non-linear problems has been recently proposed by Daversion and Prudhomme so as to alleviate the global cost of the `offline’ stage in the reduced basis method (RBM) by enriching progressively the EIM using the computed reduced basis functions. In the present work, we adapt the latter ideas to transient PDEs so as to propose an algorithm that solely requires as many full-model computations as the number of functions that span both the reduced basis and the EIM spaces. The computational cost of the procedure can therefore be substantially reduced compared to the standard strategy. Finally, we discuss possible variants of the present approach.

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Laurent Monasse: January 19 December at 3pm, A415 Inria Paris. We will present a conservative method for three-dimensional inviscid fluid-structure interaction problems. Body-fitted methods are not well-suited for large displacements or fragmentation of the structure, since they involve possibly costly remeshing of the fluid domain. We use instead an immersed boundary technique through the modification of the finite volume fluxes in the vicinity of the solid. The method is tailored to yield the exact conservation of mass, momentum and energy of the system and exhibits consistency properties. In the event of fragmentation, void can appear due to the velocity of crack opening. In order to ensure stability in the presence of void, we resort locally to the Lax-Friedrichs flux near cracks. Since both fluid and solid methods are explicit, the coupling scheme is designed to be explicit too. The computational cost of the fluid and solid methods lies mainly in the evaluation of fluxes on the fluid side and of forces and torques on the solid side. It should be noted that the coupling algorithm evaluates these only once every time step, ensuring the computational efficiency of the coupling. We also analyze a corner instability of the conservative explicit immersed boundary method in the case of a Roe flux, explain its origin and propose a way to fix the issue.

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Agnieszka Miedlar: Thursday 8 December at 3pm, A415 Inria Paris. Abstract: In the context of iterative solvers moving the eigenvalue or the eigenpair may be of particular importance in several cases, e.g., deflation techniques, increasing the spectral gap or determining the set of linearly independent eigenvectors. It can also be used for reducing the imaginary parts of the eigenvalues without chainging the matrix exponential; this can enhance the computation of $\exp(A)$. Exploiting the classical perturbation analysis for eigenvalue problems [Golub and Van Loan 2012] we study the following problem.

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Luca Formaggia: Thursday 8 December at 3pm, A415 Inria Paris. Fractures can alter greatly the characteristics of porous media. Their diverse scale distribution makes it often impossible to resort to averaging or homogenisation techniques to account for their presence. Thus, different models have been devised to account for the presence of fractures in porous media explicitly. We here present the general problem, together with a recent result of well-posedness for an hybrid dimensional mixed formulation  of Darcy flow in fractured porous media, and an analysis of a mimetic finite difference scheme adopted for its numerical solution.

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Thursday 22 September at 3pm, Jacques Louis Lions lecture hall, Inria Paris Paola ANTONIETTI: Fast solution techniques for high order Discontinuous Galerkin methods We present two-level and multigrid algorithms for the efficient solution of the linear system of equations arising from high-order discontinuous Galerkin discretizations of second-order elliptic problems. Starting from the classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the resulting multigrid schemes with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen sufficienly large.  A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented as well the extension of the proposed techniques to agglomeration-based multigrid solvers. Numerical experiments confirm the theoretical results.

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Internal seminar of the SERENA team, Thursday 11th February, 3pm-4pm in building 13: Michel Kern: Reactive transport in porous media: Formulations and numerical methods

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Internal seminar of the SERENA team, Thursday 10 December, 4pm-5pm in building 13: Nabil Birgle: Composite Method on Polygonal Meshes Abstract: We develop a reliable numerical method to approximate a flow in a porous media, modeled by an elliptic equation. The simulation is made difficult because of the strong heterogeneities of the medium, the size together with complex geometry of the domain. A regular hexahedral mesh does not allow to describe accurately the geological layers of the domain.  Consequently, this leads us to work with a mesh made of deformed cubes.  There exists several methods of type finite volumes or finite elements which solve this issue.  For our method, we wish to have only one degree of freedom per element for the pressure and one degree of freedom per face for the Darcy velocity, to stay as close to the habits of industrial software.  Since standard mixed finite element methods does not converge, our method is based on composite mixed finite element. In two dimensions, a polygonal mesh is split into triangles by adding a node to the vertices’s barycenter, and explicit formulation of the basis functions was obtained.  In dimension 3, the method extend naturally to the case of pyramidal mesh.  In the case of a hexahedron or a deformed cube, the element is divided into 24 tetrahedra by adding a node to the vertices’s barycenter and splitting the faces into 4 triangles.  The basis functions are then built by solving a discrete problem.  The proposed methods have been theoretically analyzed and completed by a posteriori estimators.  They have been tested on academical and realistic examples by using parallel computation.

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Internal seminar of the SERENA team, Thursday 03 December, 4pm-5pm in building 13: Francois Clément: Safe and Correct Programming for Scientific Computing Abstract: The increasing complexity of algorithms for modern scientific computing  makes it a major challenge to implement them in the traditional imperative languages that are popular in the community. The idea is to  explore the usage of formal tools from computing science, and in particular from the functional programming school, to design and  implement generic tools that may ease the development of scientific computing software. In this lecture, we will focus on: Sklml, an easy coarse grain parallelization compiler system; Ref-indic, a generic inversion platform for adaptive parameter  estimation; a comprehensive mechanical proof of correctness of a C program as a  PDE solver. Basic examples of the use of OCaml, Sklml, and Coq will be given.

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