### Abstracts for the Talks

We discuss several topics in the numerical
approximation of eigenvalues and eigenfunctions.
For simplicity we consider the Laplace equation although the results
are also valid for more general elliptic problems.
It is known that the standard finite element method produces
upper approximations of the eigenvalues. Therefore, a natural question
is whether it is possible to obtain lower bounds by using other methods.
We recall old results obtained in this direction in the context
of finite differences and present some results obtained by using
numerical integration and non-conforming methods.
Finally we introduce and analyze a posteriori error estimators
both for standard and non-conforming methods and present some numerical
examples.

#### Lars Grasedyck (Leipzig, Germany)

Hierarchical Matrix Approximation for the Discretisation and
Solution of Partial Differential and Integral Equations

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The efficient discretisation of (linear) integral equations
consists of three different tasks: the construction of a
suitable trial space where the solution is sought,
the computation of the entries of the stiffness matrix
--- typically involving cubature formulae for singular
integrals --- and a compression of the matrix in order to avoid
the quadratic cost for the storage and setup of the
fully populated stiffness matrix.
In the first part of this talk we will focus on the third task,
namely the matrix compression. We use the hierarchical matrix
format to store the stiffness matrix in a data-sparse format
and apply an algebraic recompression technique that aims at finding
an optimal format for the storage and evaluation of the matrix.
This recompression technique is at the same time useful to speed
up the standard hierarchical matrix arithmetic that is used for
the solution of the discrete system.
In the second part of the talk the hierarchical matrix technique
for non-local operators is applied to decompose the sparse
Galerkin stiffness matrix of a second order elliptic operator
into a product of triangular matrices. The triangular factors
are stored in the hierarchical matrix format so that the
decomposition and solution of the triangular systems can be done
in almost linear complexity.

Let f and g be functions of an hp-element space in R with bounded support.
Here, an hp-element space is characterised by local grid sizes h_l=2^{-l}*h
(l: refinement level). On each subinterval the functions are polynomials
of degree \leq p (no continuity between the subintervals required). The
convolution f*g is the integral of f(y)g(x-y)dy over R. Its (exact)
orthogonal L2 projection into an hp-element space is to be computed.
We describe the algorithm which has the complexity O(p2*N*log(N)).
Here, N is the number of all subintervals involved in the description of
the factors f,g and of the result, p is the maximal polynomial degree.

We analyze the Ritz-Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace. These estimates re°ect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities, thus resulting in different levels of errors for the approximate eigenvalues. For clustered eigenvalues, we derive eigenvalue error bounds that do not depend on the width of the cluster. Our results are readily applicable
to the classical Ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators.