Research Seminar SoSe 2013

Abstracts

Lucy Weggler: Advances in High Order Boundary Element Methods

The use of high order approximation methods is very effective in achieving high accuracy numerical simulations while keeping the number of unknowns moderate, in particular for piece- wise smooth solutions of partial differential equations. The numerical and theoretical studies for this kind of methods began in the late 70s - early 80s. Since that time high order discretization schemes are getting increasingly popular in many practical applications such as fluid dynamics, structural mechanics, electromagnetics, acoustics, etc. Nowadays, one can say that the high order methods are an established field of research in the finite element community [1, 2]. In what concerns the boundary element community, most recent publications show that theory and numerics of the high order boundary element methods are getting more and more interesting [3, 4, 5, 6].

This talk is concerned with the development and application of high order boundary elements as presented in [7]. After shortly recalling the theoretical results on the high order convergence rates for Galerkin solutions, the key ideas behind the high order boundary element implementation are discussed. At the one hand, this is the abstract relation between energy spaces and trace spaces that appear in variational formulations of elliptic and Maxwell boundary value problems. On the other hand, this is a general access to the definition of curved element shapes that go along well with the high order basis functions needed to discretize the variational formulations resulting from a boundary integral equation.

In the second part of this talk we consider the problem of electromagnetic scattering at the perfect electric conductor. Numerical results for the high order boundary element methods are presented. Our tests bring awareness of the necessity to enable a high order description of the geometry. This, in turn, gives rise to ongoing work on theoretical and practical tasks that come into play here, i.e., the approximation theory of finite-dimensional spaces of tangential vector fields on curved manifolds or isogeometric analysis in general.

Michael Feischl: Axioms of optimal adaptivity for FEM and BEM

We identify six axioms which cover the existing literature on optimality of adaptive algorithms. Put explicitly, if one can guarantee these six axioms to hold, the adaptive algorithm is proved to converge with the optimal rate. This abstract and problem independent approach reveals new connections, simplifications, and improvements of the existing literature on adaptive conforming, non-conforming, and mixed finite element methods, as well as adaptive boundary element methods.

Li-yeng Sung: Finite Element Methods for Fourth Order Variational Inequalities Arising from Elliptic Optimal Control Problems

In this talk we will discuss finite element methods for elliptic optimal control problems with pointwise state constraints formulated as fourth order variational inequalities. This is joint work with S. Brenner, C. Davis and Y. Zhang.

Daya Reddy: Discontinuous Galerkin Methods for Elasticity using Quadrilaterals

Various effective Discontinuous Galerkin (DG) methods have been developed for problems in elasticity, and have been shown to be uniformly convergent in the incompressible limit. All of the analyses are for meshes of simplicial elements. It will be shown through that the use of low-order quadrilateral elements leads to poor results. Some remedies are proposed and analysed: these include the use of selective reduced integration, and of polynomial approximations of degree 1 (that is, linear as opposed to bilinear). The resulting methods are shown to be uniformly convergent. Numerical examples illustrate the behaviour of the new approaches.