Lorenzo Mascotto: Thursday 29 April at 11:00 am ABSTRACT: Solutions to elliptic partial differential equations (PDEs) on smooth domains with smooth data are smooth. However, when solving a PDE on a Lipschitz polygon, its solution is singular at the vertices of the domain. The singular behaviour is known a priori: the solution can be split as a sum of a smooth term, plus series of singular terms that belong to the kernel of the differential operator appearing in the PDE. The virtual element method (VEM) is a generalization of the finite element method (FEM) to polygonal/polyhedral meshes and is based on approximation spaces consisting of solutions to local problems mimicking the target PDE and has been recently generalized to the extended VEM (XVEM). Here, the approximation spaces are enriched with suitable singular functions. A partition of unity (PU) is used to patch local spaces. The approach of the XVEM is close to that of the extended FEM. In this talk, we present a new paradigm for enriching virtual element (VE) spaces. Instead of adding special functions to the global space and eventually patch local spaces with a PU, we modify the definition of the local spaces by tuning the boundary conditions of local problems. By doing this, local VE spaces contain the desired singular functions, but there is no need to patch them with a PU. This results in an effective and slight modification of the already existing implementation of VE codes, as well as in a natural extension for existing theoretical results.

Continue reading

André Harnist: Thursday 1 April at 11:00 am Abstract: We consider Hybrid High-Order (HHO) approximations of Leray-Lions problems set in W^(1,p) with p in (1,2]. For this class of problems, negative powers of the gradient of the solution can appear in the flux. Depending on the expression of the latter, this can lead to a degeneracy of the problem when the gradient of the solution vanishes or becomes large. The goal of this presentation is to derive novel error estimates depending on the degeneracy of the problem inspired by [1,2,3]. Specifically, we show that, for the globally non-degenerate case, the energy-norm of the error has a convergence rate of (k+1), with k denoting the degree of the HHO approximation. In the globally degenerate case, on the other hand, the energy-norm of the error converges with a rate of (k+1)(p-1), coherently with the estimate originally proved in [4]. We additionally introduce, for each mesh element, a dimensionless number that captures the local degeneracy of the model and identifies the contribution of the element to the global error: from the fully degenerate regime, corresponding to a contribution in (k+1)(p-1), to the non-degenerate regime, corresponding to a contribution in (k+1), through all intermediate regimes. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments. [1] M. Botti, D. Castanon Quiroz, D. A. Di Pietro and A. Harnist, A Hybrid High-Order method for creeping flows of non-Newtonian fluids, Submitted. 2020. URL: https://hal.archives-ouvertes.fr/hal-02519233. [2] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes. Modelling, Simulation and Application 19. Springer International Publishing, 2020. ISBN: 978-3-030-37202-6 (Hardcover) 978-3-030-37203-3 (eBook). DOI: 10.1007/978-3-030-37203-3. [3] D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp., volume 86, 2017, number 307, pages 2159–2191,…

Continue reading

Omar Duran: Thursday 11 March at 15:00 There are two main approaches to derive a fully discrete method for solving the second-order acoustic wave equation in the time regime. One is to discretize directly the second-order time derivative and the Laplacian operator in space. The other approach is to transform the second-order equation into a first-order hyperbolic system. Firstly, we consider the time second-order form. We devise, analyze the energy-conservation properties, and evaluate numerically a hybrid high-order (HHO) scheme for the space discretization combined with a Newmark-like time-marching scheme. The HHO method uses as discrete unknowns cell- and face-based polynomials of some order 0 ≤ k, yielding for steady problems optimal convergence of order (k + 1) in the energy norm [1]. Secondly, inspired by ideas presented in [2] for hybridizable discontinuous Galerkin (HDG) method and the link between HDG and HHO methods in the steady case [3], first-order explicit or implicit time-marching schemes combined with the HHO method for space discretization are considered. We discuss the selection of the stabilization term and energy conservation and present numerical examples. Extension to the unfitted meshes is contemplated for the acoustic wave equation. We observe that the unfitted approach combined with local cell agglomeration leads to a comparable CFL condition as when using fitted meshes [4]. [1] D.A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics. 14 (2014) 461-472. [2] M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Computer Methods in Applied Mechanics and Engineering, 300:748–769, March 2016. [3] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modelling and…

Continue reading

Buyang Li: Thursday 25 February at 14:00 We prove that for a given smooth initial value, if one finite element solution of the three-dimensional time-dependent Navier–Stokes equations is bounded by $M$ when some sufficiently small step size $\tau < \tau _M$ and mesh size $h < h_M$ are used, then the true solution of the Navier–Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.

Continue reading

18 February – Roland Maier Wave propagation in heterogeneous and nonlinear media has arisen growing interest in the last years since corresponding materials can lead to unusual and interesting effects and therefore come with a wide range of applications. An important example of such materials is Kerr-type media, where the intensity of a wave directly influences the refractive index. In the time-harmonic regime, this effect can be modeled with a nonlinear Helmholtz equation. If underlying material coefficients are highly oscillatory on a microscopic scale, the numerical approximation of corresponding solutions can be a delicate task. In this talk, a multiscale technique is presented that allows one to deal with microscopic coefficients in a nonlinear Helmholtz equation without the need for global fine-scale computations. The method is based on an iterative and adaptive construction of appropriate multiscale spaces based on the multiscale method known as Localized Orthogonal Decomposition, which works under minimal structural assumptions. This talk is based on joint work with Barbara Verfürth (KIT, Karlsruhe)  

Continue reading

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.

Continue reading

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.

Continue reading

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.

Continue reading

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.

Continue reading