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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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 19 November, 4pm-5pm in building 13: Géraldine Pichot: Generation algorithms of stationary Gaussian random fields Abstract: Flow and transport equation in geological media involve physical coefficients like the permeability and the porosity. Those coefficients are classically modeled by Gaussian random fields. In order to study the impact of the variability of those coefficients on flow and transport phenomenon, a large number of those fields are simulated which requires an efficient simulation method. I will present different algorithms to simulate stationary Gaussian random fields over a regular grid and based on the classical circulant embedding approach. I will also discuss some parallel issues and present numerical results with different covariance functions. This is a joint work with Jocelyne Erhel and Mestapha Oumouni (INRIA, Rennes).

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Internal seminar of the SERENA team, Thursday 5 November, 4pm-5pm in building 13: Iain Smears: Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method Abstract: The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, its practical use has been limited by the large and challenging nonsymmetric systems to be solved at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. We first construct a left preconditioner, based on inf-sup theory, that transforms the linear system to a symmetric positive definite problem that can be solved by the preconditioned conjugate gradient (PCG) algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number κ bounded by 4 for any time-step size, any approximation order and any positive self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate  preconditioners, and show the feasibility of the high-order solution of large problems.

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Internal seminar of the SERENA team, Thursday 29 Octobre, 3pm-4pm in building 13: Sarah Ali Hassan: A posteriori error estimates for domain decomposition methods Abstract: This work develops tight a posteriori error estimates for the Domain Decomposition (DD) method with Robin transmission conditions in mixed finite element discretizations. An interface problem is solved iteratively where at each iteration local subdomain problems are solved, and information is then transferred to the neighboring subdomains. By estimating the error of the DD method as well as the discretization error, an effective criterion to stop the DD iterations is developed. The a posteriori estimates are based on the reconstruction techniques for pressures and fluxes.

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