# Research Seminar WS 2019/20

## Organization [ back ]

Prof. Carsten Carstensen
Contact: Sophie Puttkammer (puttkams(at)math.hu-berlin.de)

## Location [ back ]

Humboldt-Universität zu Berlin, Institut für Mathematik
Rudower Chaussee 25, 12489 Berlin
Room 2.417 or Room 3.007

## Schedule [ back ]

Date Time Talk by Title Room
October 16, 2019 11:15 Georgi Mitsov
(HU Berlin)
On the well-posedness of a generalized dPG time-stepping methods for the heat equation 3.007
October 23, 2019 11:15 Julian Streitberger
(HU Berlin)
Two Lowest Order MFEM Examples 2.417
October 23, 2019 11:45 Sophie Puttkammer
(HU Berlin)
A modified HHO method to compute guaranteed lower eigenvalue bounds 2.417
October 29, 2019 11:15 Tim Ricken
(Universität Stuttgart)
Neue Methoden zur Modellierung von Mehrphasenmaterialien mit Mehrskalenansätzen - Anwendungsbeispiele aus den Bereichen Materialwissenschaft, Biomechanik und Umwelttechnik 2.417
October 30, 2019 11:15 Tien Tran-Ngoc
(HU Berlin)
HHO methods for a class of degenerate convex minimization problems 3.007
October 31, 2019 9:30 Max Gunzburger
(Florida State University)
Integral equation modeling for anomalous diffusion and nonlocal mechanics 2.417
November 6, 2019 11:15 Julia Schäffer
(HU Berlin)
Convergence rates for the FEAST algorithm with dPG resolvent discretization 2.417
November 6, 2019 14:45 Neela Nataraj
(IIT Bombay)
Finite element methods for nematic liquid crystals 2.417
November 13, 2019 11:15 Ornela Mulita
(SISSA)
Smoothed Adaptive Finite Element Methods 2.417
November 20, 2019 11:15 Ma Rui
(HU Berlin)
Convergence of the adaptive nonconforming element method for an obstacle problem 2.417
November 27, 2019 15:00 Stefan Sauter
(Universität Zürich)
𝒟ℋ2 Matrices and their Application to Scattering Problems 2.417
December 4, 2019 11:15 Céline Torres
(Universität Zürich)
Stability of the Helmholtz equation with highly oscillatory coefficients 2.417
December 11, 2019 11:15 Benedikt Gräßle
(HU Berlin)
Numerical Analysis with HHO methods 2.417
December 18, 2019 11:00 Henrik Schneider
(HU Berlin)
DPG for the Laplace eigenvalue problem 2.417
January 8, 2020 11:15 Sophie Puttkammer
(HU Berlin)
Remarks on optimal convergence rates for guaranteed lower eigenvalue bounds with a modified nonconforming method 2.417
January 10, 2020 13:00 Oliver Sander
(TU Dresden)
Geometric Finite Element 3.008
January 15, 2020 11:15 Philipp Bringmann
(HU Berlin)
Convergence proofs for adaptive LSFEMs and implementation of the octAFEM3D software package 2.417
January 22, 2020 11:15 Joscha Matysiak
(HU Berlin)
This presentation is rescheduled on February 12th.
January 29, 2020 11:15 Georgi Mitsov
(HU Berlin)
Well-posedness of generalized dPG methods with locally weighted test-search norms for the heat equation 3.007
January 29, 2020 16:15 Alexander Heinlein
(Universität zu Köln)
Overlapping Schwarz preconditioning techniques for nonlinear problems 2.417
January 31, 2020 13:30 Philipp Bringmann
(HU Berlin)
Adaptive least-squares finite element method with optimal convergence rates 2.417
February 12, 2020 11:15 Joscha Matysiak
(HU Berlin)
TBA TBA

## Abstracts [ back ]

Speaker: Max Gunzburger Integral equation modeling for anomalous diffusion and nonlocal mechanics We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems. Neela Nataraj Finite element methods for nematic liquid crystals We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimate in the energy norm is derived and the quadratic convergence of Newton’s iterates is illustrated. This is a joint work with Ruma Maity and Apala Majumdar. Ornela Mulita Smoothed Adaptive Finite Element Methods We propose a new algorithm for Adaptive Finite Element Methods based on Smoothing iterations (S-AFEM). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. The method reduces the overall computational cost of AFEM by providing a fast procedure for the construction of a quasi-optimal mesh sequence with large algebraic error in the intermediate cycles. Indeed, even though the intermediate solutions are far from the exact algebraic solutions, we show that their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. In this talk, we will quantify rigorously how the error propagates throughout the algorithm, and then we will provide a connection with classical a posteriori error analysis. Finally, we will present a series of numerical experiments that highlights the efficiency and the computational speedup of S-AFEM. Stefan Sauter 𝒟ℋ2 Matrices and their Application to Scattering Problems The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our talk we will generalize the directional ℋ2 -matrix techniques from the "pure" Helmholtz operator $Lu=-\Delta u+{z}^{2}u$ with $z=-ik;\phantom{\rule{0.222em}{0ex}}k$ real, to general complex frequencies $z$ with $Re\left(z\right)>0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contain $Re\left(z\right)$ and $Im\left(z\right)$ in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We present an error analysis which is explicit with respect to the expansion order and with respect to the real and imaginary part of $z$. This allows us to choose the variable expansion order in a quasi-optimal way depending on $Re\left(z\right)$ but independent of, possibly large, $Im\left(z\right)$. The complexity analysis is explicit with respect to $Re\left(z\right)$ and $Im\left(z\right)$ and shows how higher values of $Re\left(z\right)$ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation. This talk comprises joint work with S. Börm, Christian-Albrechts-Universität Kiel, Germany and M. Lopez-Fernandez, Sapienza Universita di Roma, Italy. Céline Torres Stability of the Helmholtz equation with highly oscillatory coefficients Existence and uniqueness of the heterogeneous Helmholtz problem on bounded domains can be shown using a unique continuation principle in Fredholm's alternative. This results in an energy estimate of the problem, with a stability constant that is not directly explicit in the coefficients or the wave number. We show, that for highly oscillatory coefficients, the solution can exhibit a localised wave. As a result the stability grows exponentially in the wave number. We discuss how this constant enters in the condition for quasi-optimality, when using a hp-Finite Element Method to discretise the problem and the difficulties that arise therein.