Category: News

Quang Duc Bui: Thursday 1 June at 11:30am, A415 Inria Paris. Parareal method is a numerical method to solve time-evolution problems in parallel, which uses two propagators: the coarse – fast and inaccurate – and the fine – slow but more accurate. Instead of running the fine propagator on the whole time interval, we divide the time space into small time intervals, where we can run the fine propagator in parallel to obtain the desired solution, with the help of the coarse propagator and through parareal steps. Furthermore, each local subproblem can be solved by an iterative method, and instead of doing this local iterative method until convergence, one may perform only a few iterations of it, during parareal iterations. Propagators then become much cheaper but sharply lose their accuracy, and we hope that the convergence will be achieved across parareal iterations. Here, we propose to couple Parareal with a well-known iterative method – Schwarz Waveform Relaxation (SWR)- with only few SWR iterations in the fine propagator and with a simple coarse propagator deduced from Backward Euler method. We present the analysis of this coupled method for 1-dimensional advection reaction diffusion equation, for this case the convergence is at least linear. We also give some numerical illustrations for 1D and 2D parabolic equations, which shows that the convergence is much faster in practice.

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Quanling Deng: Tuesday 16 May at 3 pm, A415 Inria Paris. The isogeometric analysis (IgA) is a powerful numerical tool that unifies the finite element analysis (FEA) and computer-aided design (CAD). Under the framework of FEA, IgA uses as basis functions those employed in CAD, which are capable of exactly represent various complex geometries. These basis functions are called the B-Splines or more generally the Non-Uniform Rational B-Splines (NURBS) and they lead to an approximation which may have global continuity of order up to $p-1$, where $p$ is the order of the underlying polynomial, which in return delivers more robustness and higher accuracy than that of finite elements. We apply IgA to wave propagation and structural vibration problems to study their dispersion and spectrum properties. The dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. By blending optimally two standard Gaussian quadrature schemes for the integrals corresponding to the stiffness and mass, the dispersion error of IgA is minimized. The blending schemes yield two extra orders of convergence (superconvergence) in the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. To analyze the eigenvalue and eigenfunction errors, the Pythagorean eigenvalue theorem (Strang and Fix, 1973) is generalized to establish an equality among the eigenvalue, eigenfunction (in L2 and energy norms), and quadrature errors.

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Sarah Ali Hassan: Thursday 11 May at 3 pm, A415 Inria Paris. In this work we develop a posteriori error estimates and stopping criteria for domain decomposition (DD) methods with optimized Robin transmission conditions on the interface. Steady diffusion equation using the mixed finite element (MFE) discretization as well as in the heat equation using the MFE method in space and the discontinuous Galerkin scheme in time are analysed. For the heat equation, a global-in-time domain decomposition method is used, allowing for different time steps in different subdomains. We bound the error between the exact solution of the PDE and the approximate numerical solution at each iteration of the domain decomposition algorithm. Different error components (domain decomposition, space discretization, time discretization) are distinguished, which allows us to define efficient stopping criteria for the DD algorithm. The estimates are based on the reconstruction techniques for pressures and fluxes. Numerical experiments illustrate the theoretical findings.

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Ivan Yotov: Tuesday 6 June at 11 am, A415 Inria Paris. We discuss mixed finite element approximations for the Biot system of poroelasticity. We employ a weak stress symmetry elasticity formulation with three fields – stress, displacement, and rotation, as well as a mixed velocity-pressure Darcy formulation. The method is reduced to a cell-centered scheme for the displacement and the pressure, using the multipoint flux mixed finite element method for flow and the recently developed multipoint stress mixed finite element method for elasticity. The methods utilize the Brezzi-Douglas-Marini spaces for velocity and stress and a trapezoidal-type quadrature rule for integrals involving velocity, stress, and rotation, which allows for local flux, stress, and rotation elimination. We perform stability and error analysis and present numerical experiments illustrating the convergence of the method and its performance for modeling flows in deformable reservoirs. This is joint work with Ilona Ambartsumyan and Eldar Khattatov, University of Pittsburgh.

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Jannelle Hammond: Thursday 13 April at 15 pm, A315 Inria Paris. With increased pollutant emissions and exposure due to mass urbanization worldwide, air quality measurement campaigns and epidemiology studies on air pollution and health effects have become increasingly common to estimate individual exposures and evaluate their association to various illnesses. As air pollution concentrations are known to be highly heterogeneous, sophisticated physically based air quality models (AQMs), in particular CFD based models, can provide spatially rich approximations and enable to better estimate individual exposure. In this work we investigate reduced basis (RB) methods [1] to diminish the resolution cost of advanced AQMs developed for concentration evaluation at urban scales. These models depend on varying parameters including meteorological conditions and pollutant emissions, often unknown at the micro scale. RB methods use approximation spaces made of suitable samples of solutions of AQMs governed by parameterized partial differential equations (PDEs), to rapidly construct accurate and computationally efficient approximations. A key to this technique is decomposing computational work into an offline and online stage. The RB functions used to build approximation spaces and all expensive parameter-independent terms, are computed “offline” once and stored, whereas inexpensive parameter-dependent quantities are evaluated “online “ for each new value of the parameters. However, the decomposition of the matrices into offline-online pieces requires modifying the calculation code, an intrusive procedure, which in some situations is impractical. In this work, we extend the Parameterized-Background Data-Weak (PBDW) method introduced in [2] to physically based AQMs. We will generate a sample of solutions from physical AQMs with varying meteorological conditions and pollution emissionsto build the RB approximation space and combine it with experimental observations, using the method in [3], to improve pollutant concentration estimations, with the goal of collaboration with an epidemiology exposure assessment team at the University of California-Berkeley. The goal…

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Mohammad Zakerzadeh: Thursday 30 March at 10 am, A415 Inria Paris. The well-posedness of the entropy weak solutions for scalar conservation laws is a classical result. However, for multidimensional hyperbolic systems, some theoretical and numerical evidence cast doubt on that entropy solutions constitute the appropriate solution paradigm, and it has been conjectured that the more general EMV solutions ought to be considered the appropriate notion of solution. In the numerical framework and building on previous results, we prove that bounded solutions of a certain class of space-time discontinuous Galerkin (DG) schemes converge to an EMV solution. The novelty in our work is that no streamline-diffusion (SD) terms are used for stabilization. While SD stabilization is often included in the analysis of DG schemes, it is not commonly found in practical implementations. We show that a properly chosen nonlinear shock-capturing operator success to provide the necessary stability and entropy consistency estimates. In case of scalar equations this result can be strengthened and the reduction to the entropy weak solution is obtained. We prove the boundedness of the solutions as well as the consistency with all entropy inequalities, and consequently the convergence to the entropy weak solution is obtained. For viscous conservation laws, we extend our framework to general convection-diffusion systems, with both nonlinear convection and nonlinear diffusion, such that the entropy stability of the scheme is preserved. It is well-known that this property is not guaranteed, even if the convective discretization is entropy stable with respect to the purely hyperbolic problem with a naive formulation of the viscous fluxes. We use a mixed formulation, and handle the difficulties arising from the nonlinearity of the viscous flux by an additional Galerkin projection operator. We prove the entropy stability of the method for different treatments of the viscous flux, thus unifying and extending…

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Karol Cascavita: Thursday 23 March at 3 pm, A415 Inria Paris. Bingham fluids model are a group of non-Newtonian fluids with a wide and diverse range of applications in industry and research. These materials are governed by a yield limit stress, which determines solid- or fluid-like features. This behavior is model by a viscoplastic term that introduces a non-linearity in the constitutive equations. Hence, the great difficulty to solve the problem, due to the non-regularity along with the a priori unknown solid-fluid boundaries. The yield stress model considered is the Bingham model, which despite being the simplest viscoplastic model is still considered a hot problem to solve theoretically and experimentally. The approaches proposed to handle this difficulties are mainly regularization methods and augmented Lagrangian algorithms. The first technique adds a regularization parameter to smooth the problem avoiding the singularity in the rigid zones. This procedure permits an straightforward implementation at the expense of a deterioration on the accuracy. The remaining technique solves the variational problem by uncoupling nonlinearities and the gradients. All the above methods are mainly approximating solutions in a finite-element or a finite volume framework. In this work, we focus on a different discretization technique named the Discontinuous Skeletal method, introduced recently by Di Pietro et al. The aim of this work is to perform an h-adaptation to enhance the prediction of the solid-liquid boundary, exploding the salient features of the DISK method. For instance: supports general meshes, face and cell-based unknowns formulation, high-order reconstruction operator, locally conservative.

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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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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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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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