Program

Timeline for Monday, April 15th

The presentations take place in the Humboldt-Kabinett (RUD 25).

09:15	Opening
09:30	S. Sauter	Finite Elements for Elliptic Eigenvalue Problems in the Preasymptotic Regime
10:30	J. Gedicke	Benchmark Computation of Eigenvalues with Large Defect for Non-Selfadjoint Elliptic Differential Operators
11:00	Coffee break
11:30	D. Peterseim	Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
12:30	Lunch break
14:15	T. Vejchodsky	Fully computable error bounds for eigenfunctions
15:15	M. Vohralik	Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters
16:15	Coffee break
16:45	J. Storn	Computation of the LBB constant with a least-squares finite element method
17:15	R. Ma	Guaranteed lower bounds for eigenvalues of elliptic operators in any dimension
17:30	D. Boffi (TBC) F. Bertrand	A posteriori error analysis for the mixed Laplace eigenvalue problem
18:15	Closing
19:15	Dinner in Berlin Mitte

Timeline for Tuesday, April 16th

There will be informal discussions and individual appointments from 9:15 to 17:15 in Adlershof.

Abstracts

Speaker:	S. Sauter
Title:	Finite Elements for Elliptic Eigenvalue Problems in the Preasymptotic Regime
Convergence rates for finite element discretisations of elliptic eigenvalue problems in the literature usually are of the form: If the mesh width h is fine enough then the eigenvalues resp. eigenfunctions converge at some well-defined rate. In our talk, we will analyse the maximal mesh width h0 - more precisely the minimal dimension of a finite element space - so that the asymptotic convergence estimates hold for h<h0. This mesh width will depend on the size and spacing of the exact eigenvalues, the spatial dimension and the local polynomial degree of the finite element space. We will show the results of some numerical experiments concerning a) the convergence of the eigenfunctions and - values, b) the convergence of the eigenvalue multigrid method to investigate the sharpness of the theoretical results. This talk comprises joint work with L. Banjai and S. Börm.
Speaker:	J. Gedicke
Title:	Benchmark Computation of Eigenvalues with Large Defect for Non-Selfadjoint Elliptic Differential Operators
In this talk we present benchmark problems for non-selfadjoint elliptic eigenvalue problems with large defect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficient and mixed boundary conditions. Numerical experiments are performed to investigate the convergence of a Galerkin finite element method with respect to the discretization parameters, the regularity of the problem, and the ascent of the eigenvalue. This allows us to verify the sharpness of the theoretical estimates from the literature with respect to these parameters. We provide numerical evidence about the size of the ascent and show that it is important to consider the mean value for the eigenvalue approximation.
Speaker:	D. Peterseim
Title:	Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
Reference:	arxiv.org/abs/1812.00835
We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii (nonlinear Schrödinger) eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges exponentially fast to an eigenfunction. Its forward Euler time discretization yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem. This is joint work with Patrick Henning (KTH Stockholm).
Speaker:	T. Vejchodsky
Title:	Fully computable error bounds for eigenfunctions
For the Laplace eigenvalue problem we use two-sided a posteriori error bounds on eigenvalues and derive fully computable error bounds for eigenfunctions. Estimation of eigenfunctions is a delicate problem, because individual eigenfunctions do not depend continuously on problem data, in general. Eigenfunctions corresponding to tight clusters of eigenvalues are sensitive to small perturbations of the problem and eigenfunctions corresponding to multiple eigenvalues exhibit even a discontinuous behaviour. To overcome this problem, we consider spaces of eigenfunctions corresponding to clusters of eigenvalues. We derive a guaranteed bound on the directed distance between the exact and approximate space of eigenfunctions. In the case of simple eigenvalues it simplifies to the bound from [G. Birkhoff, C. De Boor, B. Swartz, B. Wendroff, SIAM J. Numer. Anal. 3 (1966), 188-203] and it can be used to estimate the usual energy norm of the difference between the exact and approximate eigenfunction. This is joint work with Xuefeng Liu.
Speaker:	M. Vohralik
Title:	Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters
We present a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors given through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schroedinger operator with periodic boundary conditions of the form -Laplace + V discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.
Speaker:	J. Storn
Title:	Computation of the LBB constant with a least-squares finite element method
An investigation of a least-squares finite element method (LSFEM) for the Stokes problem reveals a relation of the Ladyzhenskaya-Babuška-Brezzi (LBB) constant and the ellipticity constants of the LSFEM. While the approximation of the LBB constant with standard numerical methods is very challenging, the approximation of the ellipticity constants is in a Rayleigh-Ritz-like environment. This setting is well-understood and so leads to an easy to implement convergent numerical scheme for the computation of the LBB constant.
Speaker:	R. Ma
Title:	Guaranteed lower bounds for eigenvalues of elliptic operators in any dimension
In this talk, we introduce a novel generalized Crouzeix–Raviart element. This new element can produce asymptotic lower bounds for eigenvalues of general second order elliptic operators, and guaranteed lower bounds by a simple post-processing method.
Speaker:	D. Boffi (TBC)
Title:	A posteriori error analysis for the mixed Laplace eigenvalue problem
This talk considers the homogenous Laplace eigenvalue problem on polygonal domains and in mixed formulation. The problem is discretized using Raviart-Thomas finite elements of arbitrary polynomial degree on adaptive meshes. Using the Prager-Synge hypercircle approach with local flux reconstructions, a fully computable upper bound for the flux error in the L2-norm is derived, provided that the eigenvalue is simple. Efficiency of the local error estimators is proven and numerical experiments where convergence rates are studied will be provided. This is joint work with Fleurianne Bertrand and Rolf Stenberg.