Research Seminar
SS 2019

Table of Contents


Organization [ back ]

Prof. Carsten Carstensen
Contact: Friederike Hellwig (hellwigf@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 15-16, 2019 All day Various speakers Eigenvalue Day 2019 3.143A
April 24, 2019 9:15 Tien Tran-Ngoc
(HU Berlin) 		An adaptive lowest order dPG-FEM for the Stokes equations with optimal convergence rate 3.007
May 8, 2019 9:15 Philipp Bringmann
(HU Berlin) 		H¹ vector potentials for solenoidal vector fields with partial boundary conditions 3.007
May 15, 2019 9:15 Friederike Hellwig
(HU Berlin) 		An adaptive lowest order dPG-FEM for linear elasticity 3.007
May 22, 2019 9:15 Fleurianne Bertrand,
Carsten Carstensen
(HU Berlin)		 A posteriori error estimation for HHO-methods 3.007
May 29, 2019 9:15 Dietmar Gallistl
(University of Twente) 		Taylor-Hood discretization of the Reissner-Mindlin plate 3.007
May 31, 2019 17:15 Michael Feischl
(TU Wien) 		Baysian FEM 2.416
June 5, 2019 9:15 Fleurianne Bertrand,
Carsten Carstensen
(HU Berlin) 		A posteriori error estimation for HHO-methods - Part II 3.007
June 6, 2019 9:15 Devika Shylaja
(IIT Bombay) 		The Hessian Discretisation Method for fourth order elliptic equations 2.417
June 6, 2019 13:15 Gaddam Sharat
(IIT Bombay) 		Two New Approaches for Solving Elliptic Obstacle Problems Using Discontinuous Galerkin Methods 2.417
June 12, 2019 9:15 Derk Frerichs
(HU Berlin)		 On pressure robustness and adaptivity of a Virtual Element Method for the Stokes problem 2.417
June 13, 2019 9:15 Mauro Perego Generalized Moving Least Squares: Approximation Theory and Applications 2.417
June 19, 2019 9:15 Gaddam Sharat
(IIT Bombay) 		A Posteriori Error Estimates of Discontinuous Galerkin Methods for the Elliptic Obstacle Problem 2.417
June 26, 2019 9:15 Neela Nataraj
(IIT Bombay)		 Morley FEM for a distributed optimal control problem governed by the von Karman equations 2.417
July 5, 2019 9:15 Johannes Storn
(HU Berlin) 		Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis 2.417
July 10, 2019 9:15 Julia Schaeffer
(HU Berlin) 		FEAST spectral approximation using dPG resolvent discretization 2.417
July 11, 2019 16:00 Amiya Pani
(IIT Bombay) 		On Fractional in Time Evolution Problems: Some Theoretical and Computational Studies 2.417

Abstracts [ back ]

Speaker: Devika Shylaja
Title: The Hessian Discretisation Method for fourth order elliptic equations
Fourth order elliptic partial differential equations appear in various domains of mechanics. They model for example thin plates deformations as well as 2D turbulent flows through the vorticity formulation of Navier-Stokes equations. Many numerical methods, most of them are finite elements, have been developed over the years to approximate the solutions of these models. In this talk, we will present the Hessian Discretisation Method (HDM), a generic analysis framework that encompasses many numerical methods for fourth-order problems: conforming and nonconforming finite element methods, methods based on gradient recovery operators, and finite volume-based schemes. The principle of the HDM is to describe a numerical method using a set of four discrete objects, together called a Hessian Discretisation (HD): the space of unknowns, and three operators reconstructing respectively a function, a gradient and a Hessian. Each choice of HD corresponds to a specific numerical scheme. The beauty of the HDM framework is to identify four model-independent properties on an HD that ensure that the corresponding scheme converges for a variety of models, linear as well as non-linear.
Speaker: Gaddam Sharat
Title: Two New Approaches for Solving Elliptic Obstacle Problems Using Discontinuous Galerkin Methods
The main aim of the talk is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods. In the talk, using the localized behaviour of DG methods, we discuss an optimal order(with respect to regularity) {\em a priori} error estimates, in 3 dimensions. We consider two different discrete sets, one with integral constraints and the other with nodal constraints at quadrature points. The analysis is carried out in a unified setting which holds true for several DG methods using quadratic polynomials.
Speaker: Derk Frerichs
Title: On pressure robustness and adaptivity of a Virtual Element Method for the Stokes problem
For the Stokes problem, many of the standard finite element method, e.g., Taylor-Hood, and also the virtual element method (VEM) proposed in [1] fail when it comes to small viscosity parameters ν or when the continuous pressure is complicated. In this talk, a modification of the VEM is presented which makes the method pressure robust, i.e., locking free for very small ν. For this purpose, a standard interpolation into Raviart-Thomas spaces of order k-1 will be employed which allows for a pressure robust modification of the discrete right hand side. In addition, an reliable error estimator is presented that makes adaptive mesh refinement possible. The presented numerical results will round up the presentation.
[1] L. B. da Veiga, C. Lovadina, G. Vacca: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: M2AN, 51(2). 2017
Speaker: Mauro Perego
Title: Generalized Moving Least Squares: Approximation Theory and Applications
(joint work with P. Bochev, P. Kuberry, N. Trask)
In this talk we present existence and approximation results for the reconstruction of a few classes of linear functionals, including differential and integral functionals, using the Generalized Moving Least Square (GMLS) method. These results extend or specialize classical MLS theoretical results, and rely both on the classic approximation theory for finite elements and on existence/approximation results for scattered data. In particular, we will consider the reconstruction of vector fields in Sobolev spaces and the reconstruction of differential k-forms. We show how these results can be applied to data transfer problems and to design collocation and variational meshless schemes for the solution of partial differential equations.
Speaker: Gaddam Sharat
Title: A Posteriori Error Estimates of Discontinuous Galerkin Methods for the Elliptic Obstacle Problem
This is a continuation of the previous talk titled "Two New Approaches for Solving Elliptic Obstacle Problems Using Discontinuous Galerkin Methods". In this talk, we will construct the discrete Lagrange Multipliers for both the methods(Integral constraints method and Quadrature point constraints method) and also derive the optimal order(with respect to regularity) a priori error estimates. Later part of the talk focuses upon a posteriori error estimates where we will construct error estimators for both the methods and we will have a look at the reliability of the estimators. Finally, we conclude the talk by presenting the numerical experiments checking the reliability and efficiency of the estimators.
Speaker: Neela Nataraj
Title: Morley FEM for a distributed optimal control problem governed by the von Karman equations
In this talk, we consider the distributed optimal control problem governed by the von Karman equations that describe the deflection of very thin plates defined on a polygonal domain of ℝ² with box constraints on the control variable. The talk discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower order norms for the state and adjoint variables are derived. The lower order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained. This is a joint work with Sudipto Chowdhury and Devika Shylaja.
Speaker: Johannes Storn
Title: Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis
The first part of this thesis defense explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this talk introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound.

The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuska-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this talk utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant.

The third part of the talk investigates the DPG method. This investigation relates the DPG method with the LSFEM. Hence, the results from the first part of this talk extend to the DPG method. This enables precise investigations of existing and the design of novel DPG schemes.
Speaker: Amiya Pani
Title: On Fractional in Time Evolution Problems: Some Theoretical and Computational Studies
This talk starts with some basic notions on fractional order integral operators which help to define fractional order derivatives with properties. After discussing linear time fractional ODEs, I shall move to time fractional evolution problems and try to bring out some salient features of time fractional diffusion versus heat equation. Then, spend some time on solution representation and limited smoothing property. As computational PDE is close to my heart, I finally settle with the conforming piecewise-linear finite element method (FEM) applied to approximate the solution of time-fractional diffusion equations with variable diffusivity on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution, therefore, using a delicate energy analysis, a priori optimal error bounds in L², H² , and quasi-optimal in L∞-norms are derived for the semidiscrete method for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a tm type of weights to take care of the singular behavior of the continuous solution at t=0. The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. The present analysis can be extended to other types of fractional in time evolution problems.

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