Category: News

Christoph Lehrenfeld Thursday 7th April at 11:00   ABSTRACT: Discontinuous Galerkin (DG) methods are widely used to discretize partial differential equations (PDEs) due to (a.o.) flexibility for designing robust methods and simplicity in terms of data structures. One major drawback of DG methods is, however, the increased number of (globally coupled) degrees of freedom (ndof) compared to, for instance, continuous Galerkin methods. One – by now established – remedy is the use of Hybrid DG methods. These allow reducing the globally coupled ndof essentially by introducing the concept of static condensation for DG methods. Thereby the dimensions of global linear systems that need to be solved for reduced from O(p^d) to O(p^{d−1}), where d is the space dimension and p is the polynomial degree of the finite element space. A different approach is the use of Trefftz DG methods, where a DG formulation is modified by restricting the finite element spaces to functions that element-wise solve the PDE at hand. This results in a similar reduction of globally coupled ndof, cf. Figure 1 below. However, due to several limitations, Trefftz DG methods have only been applied for special PDEs so far. On the one hand, the finite element spaces have to be specifically tailored for each PDE type, on the other hand, inhomogeneous equations and non-constant coefficients in the differential operators are difficult to deal with and rarely treated. In this talk, we introduce Embedded Trefftz DG methods which exploit the existence of an underlying standard DG formulation for an efficient and flexible implementation of Trefftz DG methods. Furthermore, we relax Trefftz DG methods in view of the constraints on the finite element space leading to weak Trefftz DG spaces. Both together allow us to remove the limitations of Trefftz DG methods in order to enable them for a large…

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Christos Xenophontos Tuesday 19th April at 14:00   ABSTRACT: We consider singularly perturbed eigenvalue problems in one-dimension, and their numerical approximation by the (standard Galerkin) Finite Element Method (FEM). These are fourth order equations, where a small parameter multiplies the highest order derivative. We will present results for an $h$ version FEM with polynomials of degree $p$ on an exponentially graded (eXp) mesh, as well as an $hp$ version FEM on the so-called Spectral Boundary Layer (SBL) mesh. For both methods, robust optimal convergence is shown for the eigenvalues and associated eigenfunctions. Numerical results, illustrating the theory, will also be presented. This is joint work with H. G. Roos.

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Miloslav Vlasak Thursday 24th March at 11:00   ABSTRACT: We will present a posteriori error estimate for higher-order time discretizations, most importantly for the discontinuous Galerkin method, cf. [1]. Rather than the presentation of the estimates themselves, the talk shall focus on the most important ideas behind and their possible application to spatial nonconforming discretizations, most importantly to the discontinuous Galerkin method again. Overall, the talk shall rather focus on open problems than on the presentation of the fully completed results. Additionally, the ideas of reconstructions by the Radau polynomials that are the core ideas in a posteriori error estimates for time discretizations can be exploited for the direct efficiency analysis of the derived estimates. This can enable tracking the dependence of the efficiency constant on the discretization polynomial degree in 1D, cf. [2]. Possible extensions of this result to multiple dimensions shall be discussed. References [1] V. Dolejsi, F. Roskovec, M. Vlasak: A posteriori error estimates for higher order space-time Galerkin discretizations of nonlinear parabolic problems, SIAM J. Numer. Anal. 59, N. 3, 1486–1509 (2021) [2] M. Vlasak: On polynomial robustness of flux reconstructions, Appl. Math. 65, N. 2, 153–172 (2020)

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‪Idrissa Niakh Thursday 14th April at 11:00   ABSTRACT: We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-spheres with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

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Ruma Maity‬‬ Thursday 10th March at 11:00   ABSTRACT: In this talk, we focus on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well, referred to as ferronematics. We discuss the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, where the re-scaled elastic constant l is inversely proportional to the domain area. The conforming finite element method is used to approximate the regular solutions of the corresponding non-linear system of partial differential equations with cubic nonlinearity and non-homogeneous Dirichlet boundary conditions. We establish the existence and local uniqueness of the discrete solutions, error estimates in the energy and L2 norms with l- discretization parameter dependency. The theoretical results are complemented by the numerical experiments on the discrete solution profiles, and the numerical convergence rates that corroborate the theoretical estimates. This talk is based on joint works with Apala Majumdar (Department of Mathematics and Statistics, University of Strathclyde, UK) and Neela Nataraj (Department of Mathematics, Indian Institute of Technology Bombay, India).

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Frédéric Lebon‬‬ Thursday 27th January at 11:00   ABSTRACT: In this talk, we will focus on the theoretical and numerical modelling of interfaces between solids (adhesion, bonding, friction, etc.). We will present a general methodology based on matched asymptotic theory to obtain families of models including the relative rigidity of the interphase (soft or hard), geometrical or material non-linearities (plasticity, curvature, finite strain, …), damage and multiphysics couplings (thermics, piezoelectricity, …).  Numerical results will be presented.

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Pierre Matalon Thursday 3rd Feburary at 11:00   ABSTRACT: We consider a second order elliptic PDE discretized by the Hybrid High-Order method, for which globally coupled unknowns are located at faces. To efficiently solve the resulting linear system, we propose a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level. The core of the algorithm lies in the design of the prolongation operator that passes information from coarse to fine faces through the reconstruction of an intermediary polynomial of higher degree on the cells. High orders are natively handled by the use of the same polynomial degree at every grid level. The proposed algorithm requires a hierarchy of polyhedral meshes such that the faces (and not only the elements) are successively coarsened. Numerical tests on homogeneous and heterogeneous diffusion problems show fast convergence, asymptotic optimality with respect to the mesh size, robustness to the polynomial order, and robustness with respect to heterogeneity of the diffusion coefficient.

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Koondanibha Mitra Thursday 13th January at 11:00   ABSTRACT: Nonlinear advection-diffusion-reaction equations are used to model various complex flow processes in porous media, and in biological systems. They also exhibit parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies resulting in the loss of regularity of the solutions. The nonlinear degenerate nature of the equations makes it challenging to provide sharp error bounds to any numerical solutions of the problem. When discretized in time, such equations result in a sequence of nonlinear degenerate elliptic problems which requires linear iterative schemes to solve. The linear iterates can be used to provide upper/lower bounds to the error, and to separate the error contributions due to linearization and discretization. However, the nonlinearity, as before, impedes the derivation of sharp error bounds in the standard error norm. In the first part of this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of such nonlinear degenerate parabolic problems. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space-time efficiency error bounds are then obtained in a standard $H^1(H^{-1})\cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, and data oscillation are identified and separated. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for realistic cases. It is shown that the estimators correctly identify the errors up to a factor of the order of unity. In the second part, using linear iterative schemes, we derive reliable, fully computable, and efficient error bounds for the finite element solution of the elliptic problem which…

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Gregor Gantner Friday 10th December at 11:00   ABSTRACT: Currently, there is a growing interest in simultaneous space-time methods for solving parabolic evolution equations. The main reasons are that, compared to classical time-marching methods, space-time methods are much better suited for massively parallel implementation, are guaranteed to give quasi-optimal approximations from the employed trial space, have the potential to drive optimally converging simultaneously space-time adaptive refinement routines, and they provide enhanced possibilities for reduced order modelling of parameter-dependent problems. On the other hand, space-time methods require more storage. This disadvantage however vanishes for problems of optimal control, for which the solution is needed simultaneously over the whole time interval anyway. While the common space-time variational formulation of a parabolic equation results in a bilinear form that is non-coercive, [1] recently proved the well-posedness of a space-time first-order system least-squares formulation of the heat equation. Least-squares formulations always correspond to a symmetric and coercive bilinear form. In particular, the Galerkin approximation from any conforming trial space exists and is a quasi-best approximation. Additionally, the least-squares functional automatically provides a reliable and efficient error estimator. In [2], we have generalized the least-squares method of [1] to general second-order parabolic PDEs with possibly inhomogeneous Dirichlet or Neumann boundary conditions. For homogeneous Dirichlet conditions, we present in this talk convergence of a standard adaptive finite element method driven by the least-squares estimator, which has also been demonstrated in [2]. The convergence analysis is applicable to a wide range of least-squares formulations for other PDEs, answering a long-standing open question in the literature. Moreover, we employ the space-time least-squares method for parameter-dependent problems as well as optimal control problems. In both cases, the coercivity of the corresponding bilinear form plays a crucial role. [1] T. Führer and M. Karkulik. Space–time least-squares finite elements for parabolic equations.…

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