Research Seminar WiSe 2014/15

Abstracts

Asha Dond: Convergence of adaptive mixed finite element method for second order elliptic problem

The talk addresses the convergence of an adaptive mixed finite element method (AMFEM) for nonsymmetric, indefinite second order elliptic problems. First we analyze a nonconforming finite element discretization which converges owing to some a priori L² error estimates under reduced regularity assumptions. An equivalence result of nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control. The main difficulties in the analysis of convergence of AMFEM are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem.

Gerhard Starke: Adaptive First-Order System Least-Squares Computations for Elastoplasticity

We study first-order system least squares formulations involving stresses and displacements as process variables in the context of elastoplasticity models of von Mises type. Under typical assumptions on the hardening law, optimal order a priori error estimates are obtained for the approximation of stress and displacement in H (div) and H1, respectively. Numerical experiments show that optimal order convergence is observed on adaptively refined triangulations based on using the least squares functional as an posteriori error estimator. We will also comment on favourable properties of direct stress approximations in terms of accuracy of momentum balance and surface forces.

Hermann Matthies: To be or not to be Intrusive? – Solution of stochastic and parametric equations

Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. Particularly in the stochastic context this “sampling” is a common procedure. But often one wants to evaluate the solution quickly for a new set of parameters where is has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. Such approximations are used in several fields, notably optimisation and uncertainty quantification, where in the last case the parameters are random variables and one deals with stochastic equations. All these methods may be seen as functional approximations — representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation methods.

Another approach is to choose a finite set of basis functions, but rather than interpolation use some other projection onto the subspace spanned by these functions. Usually this will involve minimising some norm of the difference between the true parametric solution and the approximation. Such methods are sometimes called pseudo-spectral projections, or regression solutions, or discrete projections

In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—i.e. if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods—which produce a coupled system of equations—are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case.

On the other hand, methods which try to ensure that the approximation satisfies the parametric equation as well as possible are often based on a Rayleigh-Ritz or Galerkin type of “ansatz”, which leads to a coupled system for the unknown coefficients. This is often taken as an indication that the original solver can not be used, i.e. that these methods are “intrusive”. But in many circumstances these methods may as well be used in a non-intrusive fashion. Some very effective new methods based on low-rank approximations fall in the class of “not obviously non-intrusive” methods; hence it is important to show here this may be computed non-intrusively.

Ridgway Scott: Electron correlation in van der Waals interactions

We examine a technique of Slater and Kirkwood which provides an exact resolution of the asymptotic behavior of the van der Waals attraction between two hydrogens atoms. We modify their technique to make the problem more tractable analytically and more easily solvable by numerical methods. Moreover, we prove rigorously that this approach provides an exact solution for the asymptotic electron correlation. The proof makes use of recent results that utilize the Feshbach-Schur perturbation technique. We provide visual representations of the asymptotic electron correlation (entanglement) based on the use of Laguerre approximations.

Mechthild Thalhammer: Operator splitting methods for nonlinear Schrödinger equations

In this talk, I shall primarily address the issue of favourable time integration methods for nonlinear Schrödinger equations. The scope of applications in particular includes Gross– Pitaevskii equations describing Bose–Einstein condensates and the MCTDHF equations aris- ing in electron dynamics. For the considered class of problems, a variety of contributions confirms the favourable behaviour of operator splitting methods regarding efficiency and ac- curacy. However, in the absence of an local error control, the performance of the time discreti- sation strongly depends on the experienced scientist selecting the time stepsize in advance. The use of embedded pairs or a posteriori local error estimators allows to enhance reliability and efficiency of the numerical computations. Moreover, essential tools for a stability and error analysis justifying the use of the employed time discretisations will be presented.

Stefan Sauter: Composite Finite Elements

Composite Finite Elements are a new class of finite elements for the discretization of boundary value problems with complicated structures, e.g., in the geometry of the physical object and/or in the coefficients of the differential operator and boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the geometric details and this is especially advantageous for problems on domains with complicated micro-structures. In our talk, we will introduce this discretization method for different kinds of applications such as Poisson-type equations, Lamé equation, and Stokes equation. We will analyse its convergence in an a-priori and a-posteriori way and illustrate the analysis by numerical experiments.