Research Seminar
SS 2018

Table of Contents


Organisation [ back ]

Prof. Carsten Carstensen
Contact: Johannes Storn (storn@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
April 18, 2018 09:15 Gonzalo Gonzalez de Diego, M.Sc.
(Universität Duisburg-Essen) 		Approximating the Cauchy stress tensor in hyperelasticity 2.417
April 25, 2018 09:15 Prof. Dr. Daniel Peterseim
(Universität Augsburg) 		Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials 3.007
May 2, 2018 09:15 Friederike Hellwig, M.Sc.
(Humboldt-Universität zu Berlin) 		Optimal convergence rates for adaptive lowest-order dPG methods 2.417
May 9, 2018 09:15 Johannes Storn, M.Sc.
(Humboldt-Universität zu Berlin) 		On the analysis of discontinuous Petrov-Galerkin methods 2.417
May 16, 2018 09:15 Dr. Rui Ma
(Humboldt-Universität zu Berlin) 		Guaranteed lower bounds for eigenvalues of elliptic operators in any dimension 2.417
May 25, 2018 (Friday!) 13:15 Prof. Dr. Stefan Sauter (Universität Zürich)
Estimating the effect of data simplification for elliptic PDEs 2.417
May 30, 2018 09:15 Philipp Bringmann, M.Sc. (Humboldt-Universität zu Berlin)
Quasi-optimal convergence of adaptive LSFEM in 3D 2.417
June 6, 2018 09:15 Seungchan Ko, Postgraduate Student (University of Oxford)
Finite element approximation of incompressible chemically reacting non-Newtonian fluids 2.417
June 13, 2018 10:15 Tien Tran-Ngoc, B.Sc.
(Humboldt-Universität zu Berlin) 		Non-standard Discretisation of a Class of Degenerate Convex Minimisation Problems 2.417
June 13, 2018 15:15 PD Dr. Dietmar Gallistl
(University of Twente) 		Numerical approximation of planar oblique derivative problems in nondivergence form 2.417
June 19, 2018 13:30 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part I 2.006
June 20, 2018 09:15 Georgi Mitsov, M.Sc.
(Humboldt-Universität zu Berlin) 		Discontinuous Petrov-Galerkin Methods for the Time-Dependent Maxwell Equations 2.417
June 20, 2018 13:30 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part II 2.006
June 26, 2018 13:30 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part III 2.006
June 27, 2018 09:15 Sophie Puttkammer, M.Sc.
(Humboldt-Universität zu Berlin) 		A local a posteriori approximation error estimate for the companion operator 2.417
June 27, 2018 13:30 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part IV 2.006
June 10, 2018 13:30 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part V 2.006
June 11, 2018 9:15 Lukas Gehring, B.Sc.
(Humboldt-Universität zu Berlin) 		Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part VI 3.007

Abstracts [ back ]

Speaker: Gonzalo Gonzalez de Diego
Title: Approximating the Cauchy stress tensor in hyperelasticity
In this presentation, a least squares finite element method (LSFEM) is presented for the first order system of hyperelasticity defined over the deformed configuration in order to approximate the Cauchy stress tensor. Unlike the first Piola-Kirchhoff stress tensor, the Cauchy stress tensor is symmetric, a property intimately related to the conservation of angular momentum. With this work, we wish to explore the possibility of imposing the symmetry of the stress tensor, strongly or weakly, in non-linear elasticity.

Firstly, an overview of a LSFEM for hyperelasticity (over the reference configuration) by (Müller et al., 2014) is presented. Secondly, we address the question of under which conditions can a first order system over the deformed configuration be considered. Then, the former LSFEM is extended to the deformed configuration; we introduce a Gauss-Newton method for solving the non-linear minimization problem and show that, under small strains and stresses, the least-squares functional represents an a posteriori error estimator. Finally, we display numerical results for two test cases. These results indicate that this LSFEM is capable of giving reliable results even when the regularity assumptions from the analysis are not satisfied.
Speaker: Daniel Peterseim
Title: Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials
This talk discusses spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, the lowermost eigenstates exhibit strong localization in the form of an exponential decay. We quantify the rate of decay in terms of geometric parameters that characterize the potential and its disorder strength. This result is based on the convergence theory of iterative solvers for linear operator equations and their optimal local preconditioning by domain decomposition techniques. By the identification of spectral gaps for certain model potentials, we are able to predict the emergence of localized states. This is joint work with R. Altmann (Augsburg) and P. Henning (Stockholm).
Speaker: Friederike Hellwig
Title: Optimal Convergence Rates for Adaptive Lowest-Order dPG Methods
The discontinuous Petrov-Galerkin methodology enjoys a built-in a posteriori error control in some computable residual term plus data approximation terms. This talk establishes an alternative error estimator, which is globally equivalent, but allows for the proof of the axioms of adaptivity and so guarantees optimal convergence rates of the associated adaptive algorithm simultaneously for the four lowest-order discontinuous Petrov-Galerkin schemes.
Speaker: Johannes Storn
Title: On the analysis of discontinuous Petrov-Galerkin methods
This talk introduces a novel approach to analyze discontinuous Petrov-Galerkin methods. The key observation is the relation of the (continuous) DPG residual with a least-squares residual. This allows to apply techniques from the analysis of least-squares finite element methods to discontinuous Petrov-Galerkin methods and so results for example in (sharp) bounds for the inf-sup constant and the asymptotic exactness of the built-in error estimate. An investigation of ultra-weak DPG method for the Helmholtz equation exemplifies the practicability of the analysis.
Speaker: Rui Ma
Title: Guaranteed lower bounds for eigenvalues of elliptic operators in any dimension
The first part of this talk introduces a novel generalized Crouzeix–Raviart element. This new element can produce asymptotic lower bounds for eigenvalues of general second order elliptic operators, and guaranteed lower bounds by a simple post-processing method. The second part of this talk introduces a unified way to estimate the explicit constants related to the L2 error estimates for the interpolation of Morley–Wang–Xu elements. Therefore, using the constants, Morley–Wang–Xu elements can produce guaranteed lower bounds for eigenvalues of 2m–th order elliptic operators in n dimensions for m ≤ n.
Speaker: Stefan Sauter
Title: Estimating the effect of data simplification for elliptic PDEs
In many cases, the numerical simulation of complicated physical phenomena consists of various modelling and discretization steps. This includes data simplification (replacing complicated coefficients by simpler ones, using dimension-reduced models, applying homogenization models) and the numerical discretization. In many cases the final approximation is constructed by combining solutions of simpler problems in a sophisticated way via a postprocessing step.

The a priori and a posteriori analysis for such types of problems typically require regularity of the simplified problems. In particular, for the a posteriori error analysis the (estimated) value of the regularity constant (of the simplified problem) with respect to a W^(1,p) norm for some p>2 is required.

In our talk, we will present model problems where the a posteriori analysis leads to such regularity problems and explain theoretical tools for their estimation.
Speaker: Seungchan Ko
Title: Finite element approximation of incompressible chemically reacting non-Newtonian fluids
We consider a system of nonlinear partial differential equations modelling the motion of an incompressible chemically reacting non-Newtonian fluid. The governing system consists of a convection-diffusion equation for the concentration and the generalized Navier-Stokes equations, where the viscosity coefficient is a power-law-type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of movable joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovskii operator, De Giorgi-type regularity theorem, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents. This is joint work with Endre Suli and Petra Pustejovska.
Speaker: Tien Tran-Ngoc
Title: Non-standard Discretisation of a Class of Degenerate Convex Minimisation Problems
Convex optimisation occurs in many areas of applied mathematics and has a wide range of practical applications. This thesis considers a class of convex minimisation problems defined by a convex energy density which is not necessarily strictly convex but satisfies a convexity control. The proposed FEM employs an easy to compute non-conforming scheme. A postprocessing of the obtained solution coincides with the solution of a mixed scheme. This relation is a generalisation of the well-known Marini’s identity from elliptic PDEs to convex minimisation problems. The convergence analysis combines the properties of both schemes, namely the a priori convergence rate of the non-conforming FEM and the efficient a posteriori error control of the mixed FEM. The results are applied to the p-Laplace problem, the optimal design problem and the scalar double well problems. Numerical experiments provide evidences that the mixed FEM does not suffer from the typical reliability-efficiency gap.
Speaker: Dietmar Gallistl
Title: Numerical approximation of planar oblique derivative problems in nondivergence form
A numerical method for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains is proposed. The numerical scheme employs a mixed formulation with piecewise affine functions on curved finite element domains. The direct approximation of the gradient of the solution turns the oblique derivative boundary condition into an oblique direction condition. A~priori and a~posteriori error estimates as well as numerical computations on uniform and adaptive meshes are provided.
Speaker: Lukas Gehring
Title: Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions
Newest Vertex Bisection (NVB) is a simple algorithm refining simplicial meshes adaptively, that is locally, i.e. it should be able to refine the mesh around some marked element (where the error is relatively large, for example) letting distant elements as they are but to retain a regular and shape-regular mesh (this is in very simple terms: to avoid hanging nodes as well as too small angles). A simple algorithm means here: If a simplex is changed, it is bisected in a way predestinated by its "own" data, there is neither coarsening nor weighing several possible refinements nor dependance of the surrounding mesh. This way all simplices which can arise form a binary tree and the triangulations which can arise are certain subtrees. After a rigorous introduction into the subject, we are going to characterise these subtrees in order to found a well-structured theory of NVB and to give a new proof of the famous theorem of Binev, Dahmen and DeVore, saying that the total number of simplices grows only linearly with the total number of marked simplices.
Speaker: Georgi Mitzov
Title: Discontinuous Petrov-Galerkin Methods for the Time-Dependent Maxwell Equations
Discontinuous Petrov-Galerkin (dPG) methods utilize ”broken” test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. This leads to new unknown variables on the interface spaces to connect the broken spaces to their unbroken counterparts.

We introduce and analyse a novel family of dPG methods for the time-dependent Maxwell equations. The schemes are based on a backward Euler time-stepping and use a primal variational formulation at each time-step. The a priori error estimates yield a method with optimal convergence in the energy-norm for general Lipschitz polyhedral domains. Numerical experiments support these theoretical results and suggest the reliability of two a posteriori error estimators.
Speaker: Sophie Puttkammer
Title: A local a posteriori approximation error estimate for the companion operator
This talk presents and proves some abstract estimates for polynomials with optimal constants. Among others, these estimates allow for a local a posteriori approximation error estimate for the companion operator with explicit constants in any space dimension. The constants depend only on the maximal number of simplices in a nodal patch. The novel estimates allows to prove discrete reliability for the Crouzeix-Raviart FEM without an additional layer of simplices.

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