November 26 – Koondanibha Mitra: A posteriori error bounds for the Richards equation

Koondanibha Mitra: Thursday 26 November at 16:00 via this link Richards equation is commonly used to model the flow of water and air through the soil, and serves as a gateway equation for multiphase flow through porous domains. It is a nonlinear advection-reaction-diffusion equation that exhibits both elliptic-parabolic and hyperbolic-parabolic kind of degeneracies. In this study, we provide fully computable, locally space-time efficient, and reliable a posteriori error bounds for numerical solutions of the fully degenerate Richards equation. This is achieved in a variation of the $ H^1(H^{-1})\cap L^2(L^2) \cap L^2(H^1)$ norm characterized by the minimum regularity inherited by the exact solutions. For showing global reliability, a non-local in time error estimate is derived individually for the $H^1(H^{-1})$, $L^2(L^2)$ and the $L^2(H^1)$ error components with a maximum principle and a degeneracy estimator being used for the last one. Local and global space-time efficient error bounds are obtained, and error contributors such as flux and time non-conformity, quadrature, linearisation, data oscillation, are identified and separated. The estimates hold also in space-time adaptive settings. The predictions are verified numerically and it is shown that the estimators correctly identify the errors up to a factor in the order of unity.

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November 19 – Joëlle Ferzly: Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: adaptivity and inexact resolution

Joëlle Ferzly: Thursday 19 November at 11:00 via zoom (meeting ID: 966 1837 4193 and code: zFG6Lb) We are interested in nonlinear algebraic systems with complementarity constraints stemming from numerical discretizations of nonlinear complementarity problems. The particularity is that they are nondifferentiable, so that classical linearization schemes like the Newton method cannot be applied directly. To approximate the solution of such nonlinear systems, an iterative linearization algorithm like the semismooth Newton-min can be used. We consider smoothing methods, where the nondifferentiable nonlinearity is smoothed. In particular, a smoothing Newton algorithm based on the smoothed min or Fischer-Burmeister function, and a smoothing interior-point algorithm. The corresponding linear system is approximately solved using any iterative linear algebraic solver. We derive an a posteriori error estimate that allows to distinguish the smoothing, linearization, and algebraic error components. These ingredients are then used to formulate adaptive criteria for stopping the linear and nonlinear solver. This leads us to propose an adaptive algorithm ensuring important savings in terms of the number of cumulated algebraic iterations. We apply our analysis to the system of variational inequalities describing the contact between two membranes. We will show that the proposed algorithm, in combination with the GMRES algebraic solver, is promising in comparison with other methods.

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November 5 – Zhaonan Dong: On a posteriori error estimates for non-conforming Galerkin methods

Zhaonan Dong: Thursday 5 November at 11:00 via zoom (meeting ID: 961 6781 3449 and code: g4q44P) Non-conforming Galerkin methods are very popular for the stable and accurate numerical approximation of challenging PDE problems. The term « non-conforming » refers to approximations that do not respect the continuity properties of the PDE solutions. Nonetheless, to arrive at rigorous error control via a posteriori error estimates, non-conforming methods pose a number of challenges. I will present a novel methodology for proving a posteriori error estimates for the “extreme » class (in terms of non-conformity) of discontinuous Galerkin methods in various settings. For instance, we prove a posteriori error estimates for the recent family of Galerkin methods employing the general shaped polygonal and polyhedral elements, solving an open problem in the literature. Furthermore, with the help this new idea, we prove new a posteriori error bounds for various $hp$-version non-conforming FEMs for fourth-order elliptic problems; these results also solve a number of open questions in the literature, yet they arise relatively easily within the new reconstruction framework of proof. These results open a door to design new reliable adaptive algorithms for solving the problems in thin plate theories of elasticity, phase-field modeling and mathematical biology.

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October 22 – Théophile Chaumont-Frelet: A posteriori error estimates for Maxwell’s equations based on flux quasi-equilibration

Théophile Chaumont-Frelet: Thursday 22 October at 11:00, A415, Inria Paris. I will present of a novel a posteriori estimator for finite element discretizations of Maxwell’s equations. The construction hinges on a modification of the flux equilibration technique, called quasi-equilibration. The resulting estimator is inexpensive to compute and polynomial-degree-robust, which means that the reliability and efficiency constants are independent of the discretization order. I will first describe the standard flux equilibration technique for the simpler case of Poisson’s problem, and explain why it is hard to directly apply this idea to Maxwell’s equations. Then, I will present in detail the derivation of the proposed estimator through the quasi-equilibration procedure. Numerical examples highlighting the key features of the estimator will be presented, and followed by concluding remarks.

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October 15 – Florent Hédin: A hybrid high-order (HHO) method with non-matching meshes in discrete fracture networks

Florent Hédin: Thursday 15 October at 11:00, Gilles Kahn 1, Inria Paris. We are interested in efficient numerical methods for solving flow in large scale fractured networks. Fractures are ubiquitous in the subsurface. Flow in fractured rocks are of interest for many applications (water resources, geothermal applications, oil/gas extraction, nuclear waste disposal). The networks are modeled as Discrete Fractures Networks (DFN). The main challenges of such flow simulations are the uncertainty regarding the geometry and properties of the subsurface, the observed wide range of fractures length (from centimeters to kilometers) and the number of fractures (from thousands to millions of fractures). In natural rocks, flow is highly channelled, which motivates to mesh finely the fractures that carry most of the flow, and coarsely the remaining fractures. But independent triangular mesh generation from one fracture to another yields non matching triangles at the intersections between fractures. Mortar methods have been developed in the past years to deal with non matching grids. In this presentation, we propose an alternative based on the recent HHO method which naturally handles general meshes (polygons/polyhedral) and face polynomials of order k ≥ 0. Combined with refining/coarsening strategies, we will show how the HHO method allows to save computational time in DFN flow simulations.

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Wednesday 5 – Amina Benaceur: Model reduction for nonlinear thermics and mechanics

Amina Benaceur: Wednesday 5 December at 11:00 am, A315 Inria Paris. This thesis introduces three new developments of the reduced basis method (RB) and the empirical interpolation method (EIM) for nonlinear problems. The first contribution is a new methodology, the Progressive RB-EIM (PREIM) which aims at reducing the cost of the phase during which the reduced model is constructed without compromising the accuracy of the final RB approximation. The idea is to gradually enrich the EIM approximation and the RB space, in contrast to the standard approach where both constructions are separate. The second contribution is related to the RB for variational inequalities with nonlinear constraints. We employ an RB-EIM combination to treat the nonlinear constraint. Also, we build a reduced basis for the Lagrange multipliers via a hierarchical algorithm that preserves the non-negativity of the basis vectors. We apply this strategy to elastic frictionless contact for non-matching meshes. Finally, the third contribution focuses on model reduction with data assimilation. A dedicated method has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the method to a time-dependent framework using a POD-greedy algorithm in order to build accurate reduced spaces for all the time steps. Besides, we devise a new algorithm that produces better reduced spaces while minimizing the number of measurements required for the final reduced problem.

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January 9 – Zhaonan Dong: hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Zhaonan Dong: Wednesday 9 January at 11 am, A415 Inria Paris. PDE models are often characterised by local features such as solution singularities/layers and domains with complicated boundaries. These special features make the design of accurate numerical solutions challenging, or require huge amount of computational resources. One way of achieving complexity reduction of the numerical solution for such PDE models is to design novel numerical methods which support general meshes consisting of polygonal/polyhedral elements, such that local features of the model can be resolved in efficiently by adaptive choices of such general meshes. In this talk, we will review the recently developed hp-version symmetric interior penalty discontinuous Galerkin (dG) finite element method for the numerical approximation of PDEs on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. The key feature of the proposed dG method is that the stability and hp-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Moreover, under certain practical mesh assumptions, the proposed dG method was proven to be available to incorporate very general polygonal/polyhedral elements with an arbitrary number of faces. Because of utilising general shaped elements, the dG method shows a great flexibility in designing an adaptive algorithm by refining or coarsening general polytopic elements. Especially for solving the convection-dominated problems on which boundary and interior layers may appear and need a lot of degrees of freedom to resolve. Finally, we will present several numerical examples through different classes of linear PDEs to highlight the practical performance of the proposed method.

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December 13 – Maxime Breden: An introduction to a posteriori validation techniques, illustrated on the Navier-Stokes equations

Maxime Breden: Thursday 13 December at 11 am, A415 Inria Paris. The aim of a posteriori validation techniques is to obtain mathematically rigorous and quantitative existence theorems, using numerical simulations. Given an approximate solution, the general strategy is to combine a posteriori estimates with analytical ones to apply a fixed point theorem, which then yields the existence of a true solution in an explicit neighborhood of the approximate one. In the first part of the talk, I’ll present the main ideas in more detail, and describe the general framework in which they are applicable. In the second part, I’ll then focus on a specific example and explain how to validate a posteriori periodic solutions of the Navier-Stokes equations with a Taylor-Green type of forcing. This is joint work with Jan Bouwe van den Berg, Jean-Philippe Lessard and Lennaert van Veen.

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April 16 – Simon Lemaire: An optimization-based method for the numerical approximation of sign-changing PDEs

Simon Lemaire: Thursday 16 April at 3 pm, A415 Inria Paris. We are interested in physical settings presenting an interface between a classical (positive) material and a (negative) metamaterial, in such a way that the coefficients of the model change sign in the domain. We study, in the « elliptic » case, the numerical approximation of such sign-shifting problems. We introduce a new numerical method, based on domain decomposition and optimization, that we prove to be convergent, as soon as, for a given right-hand side, the problem admits a solution that is unique. The proof of convergence does not rely on any symmetry assumption on the mesh family with respect to the sign-changing interface. In that respect, it gives a more convenient alternative to T-coercivity based approximation in the situations when the latter is applicable, whereas it constitutes a new paradigm in the situations when the latter is not. We illustrate our findings on a comprehensive set of test-cases.

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February 20 – Thirupathi Gudi: An energy space based approach for the finite element approximation of the Dirichlet boundary control problem.

Thirupathi Gudi: Tuesday 20 February at 3 pm, A415 Inria Paris. In this talk, we review some approaches for formulating the Dirichlet boundary control problem and then we present a new energy space based approach. We show that this new approach allows high regularity for both optimal control and the optimal state. Using, the optimality conditions at continuous level, we propose a finite element method for numerical solution and derive subsequent error estimates. We show some numerical experiments to illustrate the method.

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