Research Seminar WiSe 2013/14

Abstracts

Lucy Weggler: The boundary element method in one dimension

First, we consider a boundary value problem in 1D. This one dimensional problem serves us to derive the very fundamental ingredients and steps that enter the picture when solving an elliptic PDE with constant coefficients by the boundary element method. It is shown that the solving strategy translates to boundary value problems posed in two or three space dimensions. The major goal of this lecture is to introduce the basic ideas related to the boundary element method to all those who are not familiar with it.

Dr. Yin Yang: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

In this talk, We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞; norm and weighted L²-norm. The numerical examples are given to illustrate the theoretical results.

Dr. Nianyu Yi: The superconvergent cluster recovery

Finite element recovery techniques are post-processing methods that reconstruct numerical approximations from finite element solutions to obtain improved solutions. Gradient recovery technique is widely used in engineering practice for its robustness as an a posteriori error estimator, its super-convergence of the recovered derivatives, and its efficiency in implementation. We propose a new gradient recovery technique: Super-convergent cluster recovery. A linear polynomial approximation is obtained by a least-squares fitting to the finite element solution at certain sample points, which in turn gives the recovered gradient at recovering points. The recovered gradient is superconvergent. The SCR can be used as an a posteriori error estimator, which is relatively simple to implement, cheap in terms of storage and computational cost for adaptive algorithms. Some numerical examples are presented for illustrating the effectiveness of our recovery method.

Prof. Yanping Chen: Error estimates and superconvergence of mixed finite element methods for bilinear optimal control problems

In this work, we investigate error estimates and superconvergence of the bilinear elliptic optimal control problems by Raviart-Thomas mixed finite element methods. The control variable enters the state equation as a coefficient. The state and the co-state variables are approximated by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise linear or constant functions. We obtain the superconvergence property between average L² projection and the approximation of the control variable, the convergence order is h². Finally, we derive a prioti and a posteriori error estimates for the control variable and coupled state variable.

Rob Stevenson: Instance optimality of the adaptive maximum strategy

Adaptive finite element methods (AFEM's) for elliptic problems with the bulk chasing marking strategy are known to converge with the best possible rate. Practitioners usually prefer the maximum marking strategy, since it doesn't require sorting of the error indicators, and the results are less sensitive to the choice of a marking parameter. In this talk, we present a proof that AFEM with a modified maximum strategy is even instance optimal for the total error, being the sum of the energy error and the oscillation.

Christian Kreuzer: Instance optimality of an adaptive finite element method with greedy marking

Roughly speaking, an adaptive finite element method (AFEM) is called instance optimal, when it produces almost optimal approximations with respect to the degrees of freedom (DOFs). This means that for `ANY' other admissible approximation, we have
DOFs(ANY) ≤ c DOFs(AFEM) => Error(AFEM) ≤ C Error(ANY)
with fixed constants C>1>c>0. We show instance optimality of an adaptive finite element method for the total error, which is the sum of the error and the oscillation. The method uses a greedy marking, respectively maximum strategy, as well as refinement by recursive newest vertex bisection. Our approach uses new techniques based on the minimisation of the Dirichlet energy and a newly developed tree structure of the nodes of admissible triangulations.

Martin Brokate: Rate independent evolutions: Derivatives and optimal control

We present result concerning weak derivatives of the solution operators ( = hysteresis operators) of scalar rate independent evolutions. In addition, we discuss optimality conditions for related optimal control problems in the more general vector case, obtained by regularization.

Martin Kruzik: An introduction to quasistatic adhesive contact problems in delamination

We review mathematical approaches to inelastic processes on surfaces of elastic bodies. First, we consider a quasistatic and rate-independent evolution at small strains and the so-called energetic solution. Then we extend this concept of solution to take into account viscous effects, as well. Beside the theoretical treatment, numerical experiments are also presented. This talk is based on joint works with Tomas Roubicek, Christos Panagiotopoulos, and Jan Zeman.

Manfred Dobrowolski: Parallel solution of large linear systems

We propose two additive methods for the fast parallel solution of linear systems arising in the discretization of elliptic boundary value problems. The first method can be applied to the solution of discrete Poisson's equation or convection diffusion equations on orthogonal meshes. The second method is more general and is well adapted to general discretizations of elliptic systems. Both methods can be used as preconditioners or as linear iterative methods.