March 16th – Thomas Boiveau: Approximation of parabolic equations by space-time tensor methods

Thomas Boiveau: Thursday 16 March at 3 pm, A415 Inria Paris. Abstract: In numerical simulations, the reduction of computational costs is a key challenge for the development of new models and algorithms; tensor methods are widely used for this purpose. In this work, we consider parabolic equations and define a mathematical framework in order to use iterative low-rank greedy algorithms, based on the separation of the space and time variables. The problem is handled using a minimal residual formulation. We perform numerical tests to compare the proposed method with the strategies already suggested in the literature.

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March 2nd – Matteo Cicuttin: Implementation of Discontinuous Skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming.

Matteo Cicuttin: Thursday 2 March at 3 pm, A415 Inria Paris. Abstract: Discontinuous Skeletal methods are devised at the mathematical level in a dimension-independent and cell-shape-independent fashion. Their implementation, at least in principle, should conserve this feature: a single piece of code should be able to work in any space dimension and to deal with any cell shape. It is not common, however, to see software packages taking this approach. In the vast majority of the cases, the codes are capable to run only on few very specific kinds of mesh, or only in 1D or 2D or 3D. On the one hand, this can happen simply because a fully general tool is not always needed. On the other hand, the programming languages commonly used by the scientific computing community (in particular Fortran and Matlab) are not easily amenable to an implementation which is generic and efficient at the same time. The usual (and natural) approach, in conventional languages, is to have different versions of the code, for example one specialized for 1D, one for 2D and one for 3D applications, making the overall maintenance of the codes rather cumbersome. The same considerations generally apply to the handling of mesh cells with various shapes, i.e., codes written in conventional languages generally support only a limited (and set in advance) number of cell shapes. Generic programming offers a valuable tool to address the above issues: by writing the code generically, it is possible to avoid making any assumption neither on the dimension (1D, 2D, 3D) of the problem, nor on the kind of mesh. In some sense, writing generic code resembles writing pseudocode: the compiler will take care of giving the correct meaning to each basic operation. As a result, with generic programming there will be still differents versions of the…

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February 23rd – Christian Kreuzer: Quasi-optimality of discontinuous Galerkin methods for parabolic problems

Christian Kreuzer: 23 February at 10h45 am, A415 Inria Paris. It is a well known fact that inf-sup stable Galerkin discretisations of linear continuous problem provide quasi-optimal approximations in the corresponding norms. For elliptic problems, this is e.g. known as Cea’s Lemma. A priori error bounds are then typically obtained with the help of some (quasi)-interpolation. We apply this principle to parabolic problems and prove inf-sup stability and thus quasi-optimality of space and time adaptive backward Euler-Galerkin discretisations. In a second step, we define a reasonable (quasi)-interpolation operator and conclude a priori error bounds. In 1982 Dupont presented a counter example showing non-convergence of the backward Euler-Galerkin in the presence of spatial mesh changes. In this case, our bound contains an additional term, which is consistent with Dupont’s observation.

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February 23rd – Lars Diening: Linearization of the p-Poisson equation

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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March 9th – Ludovic Chamoin: Simulations multiéchelles avec MsFEM: estimation d’erreur a posteriori, stratégie adaptative, et couplage avec la reduction de modèle

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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January 26th – Amina Benaceur: An improved reduced basis method for non-linear heat transfer

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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January 19th – Laurent Monasse: A 3D conservative coupling between a compressible flow and a fragmenting structure

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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January 5th – Agnieszka Miedlar: Moving eigenvalues and eigenvectors by simple perturbations

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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December 8th – Luca Formaggia: Hybrid dimensional Darcy flow in fractured porous media, some recent results on mimetic discretization

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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