Our research seminar addresses graduate students, young researchers, and well-established experts interested in the area of numerical analysis, optimization, and scientific computing. We aim to discuss recent developments in our field, as well as Ph.D. and Master's theses.

Prof. Dr. Rüdiger Verfürth
Jun.-Prof. Dr. Markus Weimar

Winterterm 2018 / 2019

Thu, 29.11.2018 (14:30 in IB 1/103; Friedrich-Sommer-Raum):
MARKUS HANSEN (Philipps-University Marburg)
Properties of Kondratiev spaces and Besov regularity for semilinear elliptic PDEs on polyhedral domains

Kondratiev spaces are a special type of weighted Sobolev spaces, particularly suited to describe the regularity of solutions to operator equations on polygonal or polyhedral domains.
The relevance of these spaces from the point of view of numerical analysis stems from embedding assertions into Besov spaces, which allows to derive convergence rates for adaptive wavelet and Finite element methods.
In this talk we shall discuss properties of these spaces, starting with the motivation from regularity theory, followed by structural properties (embeddings, localization principles) and pointwise multiplication assertions. Finally, all those results are combined to derive new regularity results for semilinear elliptic equations.

Winterterm 2017 / 2018

Wed, 22.11.2017 (16:15 in NA 2/24):
LARS DIENING (Bielefeld University)
Linearization of the p-Poisson equation

This is a joint work with Massimo Fornasier and Maximilian Wank. In this talk we propose a iterative method to solve the non-linear p-Poisson equation. The method is derived from a relaxed energy by an alternating direction method. We are able to show algebraic convergence of the iterates to the solution. However, our numerical experiments based on finite elements indicate optimal, exponential convergence.

Summerterm 2017

Tue, 22.08.2017 (13:15 in NA 2/64):
Nichtlineare Approximationsraten und Besov-Regularität elliptischer PDEs auf Polyedergebieten

Wed, 26.07.2017 (16:15 in NA 2/64):
PETRU A. CIOICA-LICHT (University of Otago, Dunedin, New Zealand)
Stochastic Partial Differential Equations: Regularity and Approximation

Stochastic partial differential equations (SPDEs, for short) are mathematical models for evolutions in space and time, which are corrupted by noise. Although we can prove existence and uniqueness of a solution to various classes of such equations, in general, we do not have an explicit representation of this solution. Thus, in order to make those models ready to use for applications, we need efficient numerical methods for approximating their solutions. And to determine the efficiency of an approximation method, we usually need to analyse the regularity of the target object, which is, in our case, the solution of the SPDE.
The aim of this talk is to present some recent results concerning the regularity of SPDEs and to point out their relevance for the question of developing efficient numerical methods for solving these equations. Prior to that I will give a brief overview over that parts of the already established Lp-theory for SPDEs that is relevant in this context. For simplicity, we focus on the most basic example, the stochastic heat equation driven by a (cylindrical) Wiener process.

Mon, 26.06.2017 (14:15 in NA 3/24):
ROB STEVENSON (University of Amsterdam, Netherlands)
Adaptive wavelet methods for space-time variational formulations of evolutionary PDEs

Space-time discretization methods require a well-posed space-time variational formulation. Such formulations are well-known for parabolic problems. The (Navier)-Stokes equations can be viewed as a parabolic problem for the divergence-free velocities. Yet to avoid the cumbersome construction of divergence-free trial spaces, we present well-posed variational formulations for the saddle-point problem involving the pair of velocities and pressure. We discuss adaptive wavelet methods for the optimal adaptive solution of simultaneous space-time variational formulations of evolutionary PDEs. Thanks to use of tensor products of temporal and spatial wavelets, the whole time evolution problem can be solved at a complexity of solving one instance of the corresponding stationary problem.

Wed, 21.06.2017 (16:15 in NA 01/99):
Adaptive Wavelet-Methoden für Operatorgleichungen

Diese Antrittsvorlesung gibt einen Einblick in die Grundlagen moderner Methoden der numerischen Analysis zur approximativen Lösung von Operatorgleichungen, wie sie in einer Vielzahl von Modellen naturwissenschaftlich-technischer Disziplinen auftreten. Im Zentrum der Diskussion steht dabei die Beschreibung und Analyse der sogenannten adaptiven Wavelet-Galerkin-Methode. Im Zuge dessen werden insbesondere die Grundlagen adaptiver Verfahren und die damit verbundene Regularitätstheorie in Funktionenräumen vom Sobolev- und Besov-Typ vorgestellt, sowie elementare Konzepte der nichtlinearen Approximationstheorie besprochen.

Wed, 19.04.2017 (14:15 in NA 2/64):
MAX GUNZBURGER (Florida State University, Tallahassee, USA)
Integral equation modeling for nonlocal diffusion and 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 asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

Mon, 27.03.2017 (14:15 in NA 2/64):
SUSANNE C. BRENNER (Louisiana State University, Baton Rouge, USA)
C0 Interior Penalty Methods

C0 interior penalty methods are discontinuous Galerkin methods for fourth order problems that are based on standard Lagrange finite element spaces for second order problems. In this talk we will discuss the a priori and a posteriori error analyses of these methods for fourth order elliptic boundary value problems and elliptic variational inequalities.