Ani Miraçi: Thursday 10 December at 16:00 via this link with meeting ID: 950 9381 2553 and code: 077683 We consider systems of linear algebraic equations arising from discretizations of second-order elliptic partial differential equations using finite elements of arbitrary polynomial degree. First, we propose an a posteriori estimator of the algebraic error, the construction of which is linked to that of a multigrid solver without pre-smoothing and a single post-smoothing step by overlapping Schwarz methods (block-Jacobi). We prove the following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree (p-robustness); the estimator represents a two-sided p-robust bound of the algebraic error. Next, we introduce optimal step-sizes which minimize the error at each level. This makes it possible to have an explicit Pythagorean formula for the algebraic error; this in turn serves as the foundation for a simple and efficient adaptive strategy allowing to choose the number of post-smoothing steps per level. We also introduce an adaptive local smoothing strategy thanks to our efficient and localized estimator of the algebraic error by levels/patches of elements. Patches contributing more than a user-prescribed percentage to the overall error are marked via a bulk-chasing criterion. Each iteration is composed of a non-adaptive V-cycle and an adaptive V-cycle which uses local smoothing only in the marked patches. These V-cycles contract the algebraic error in a p-robust fashion. Finally, we also extend some the above results to a mixed finite elements setting in two space dimensions. A variety of numerical tests is presented to confirm our theoretical results and to illustrate the advantages of our approaches.

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Riccardo Milani: Wednesday 9 December at 16:00 via this link with meeting ID: 996 3774 0482 and code: 190673 We develop face-based Compatible Discrete Operator (CDO-Fb) schemes for the unsteady, incompressible Stokes and Navier–Stokes equations. We introduce operators discretizing the gradient, the divergence, and the convection term. It is proved that the discrete divergence operator allows one to recover a discrete inf-sup condition. Moreover, the discrete convection operator is dissipative, a paramount property for the energy balance. The scheme is first tested in the steady case on general and deformed meshes in order to highlight the flexibility and the robustness of the CDO-Fb discretization. The focus is then moved onto the time-stepping techniques. In particular, we analyze the classical monolithic approach, consisting in solving saddle-point problems, and the Artificial Compressibility (AC) method, which allows one to avoid such saddle-point systems at the cost of relaxing the mass balance. Three classic techniques for the treatment of the convection term are investigated: Picard iterations, the linearized convection and the explicit convection. Numerical results stemming from first-order and then from second-order time-schemes show that the AC method is an accurate and efficient alternative to the classical monolithic approach.

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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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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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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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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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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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