# How to calculate singularity exponents?

### Thomas Apel

#### email: apel@mathematik.tu-chemnitz.de

The stress distribution at the top of a polyhedral corner or at a crack tip has the typical ralpha-singularity. The knowledge of the singularity exponent alpha is of interest in mechanics, e.g. for investigating the growth of cracks. The exponent determines also the regularity of the stress field and, consequently, the convergence properties of numerical methods for its approximation. Therefore the calculation of alpha is of great importance.

Mathematically, the exponent alpha is an eigenvalue of a quadratic operator eigenvalue problem, in particular the one with smallest positive real part. The corresponding eigenfunction is smooth in the case of a circular cone but in general it has singular behaviour itself.

The eigenvalue problem is solved numerically by a finite element method. General approximation results are given in the talk. By interpolation theory we can specify the approximation order for the considered approximation method. Numerical examples underline these results.

By the finite element approximation, the operator eigenvalue problem is transformed to a quadratic matrix eigenvalue problem. Properties of the matrices and strategies for the solution of this eigenvalue problem are discussed at the end of the talk.

# Parallel Computations in Material Modeling

### Zdenek Bittnar

#### Czech Republic

The modern material models are often very computationaly demanding. Some of them even extremly demanding. Many engineering structures are very complex. The recent progress in computer technology allows practical use of parallel computers for the analysis of such problems. In the presentation the overview of governing equations as well as the numerical procedures used in simulation of real structures and materials will be done. The advantages of implicit and explicit schemas will be discuss. The comparison of efficiency of monolithic paralell computers and clusters based on cheap Intel technology. Windows or Linux ? Examples from composite structures, mesh generation and modeling of damage of concrete.

# Numerical Approximations of Compressible Flow

### Miloslav Feistauer

#### e-mail: feist@karlin.mff.cuni.cz

The subject matter of the paper is the numerical solution of viscous gas flow. The method is based on  the combination of upwind flux vector splitting finite volume schemes for the approximation of inviscid terms in the compressible Navier-Stokes equations with the finite element method for the discretization of viscous terms. We use the nonconforming Crouzeix - Raviart finite elements combined with the so-callsed barycentric finite volumes associated with midpoints of sides of triangular finite elements.
The combined finite volume - finite element scheme is theoretically studied on a model nonlinear scalar conservation law equation with a diffusion term. Namely, the convergence of the scheme and error estimates are analyzed.
Some results of numerical simulation of gas flows arising from industrial applications are presented.

# Adaptive Discontinuous Galerkin finite element methods for conservation equations

### Ralf Hartmann

#### email: Ralf.Hartmann@iwr.uni-heidelberg.de

We present an approach to solving conservation equations by adaptive finite elements in form of a Discontinuous Galerkin (DG) finite element method. Shock capturing is used to reduce over-shoots at shocks. This stabilization introduces artificial diffusion at places of large local residuals, but preserves conservation and Galerkin orthogonality of the DG method.

Using a global duality argument and Galerkin orthogonality, we obtain a residual-based error representation with respect to an arbitrary output functional of the solution. This results in numerically computable error estimates which are used for mesh refinement. In this way, very economical and highly localized meshes can be generated which are tailored to the efficient computation of the quantity of interest. We will report on 1+1-dimensional numerical test cases.

References:

[1] W. Bangerth and G. Kanschat. Concepts for object-oriented finite element software -- the deal.II  library. Preprint 99-43, SFB 359, Universität Heidelberg, Oct. 1999.

[2] R. Becker and R. Rannacher. A Feed--Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples, East--West J. Num. Math., 4:237-264, 1996.

[3] R. Hartmann. Adaptive FE-methods for Conservation Equations. Submitted to Eighth International Conference on Hyperbolic Problems. Theory, Numerics, Applications (HYP2000), 2000.

[4] J. Jaffre, C. Johnson, and A. Szepessy. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models and Methods in Appl. Sciences , 5:367--386, 1995.

# Nitsche type mortaring  for elliptic problems

### Bernd Heinrich

#### email: bernd.heinrich@mathematik.tu-chemnitz.de

The Nitsche type finite element mortaring follows some idea of Nitsche (1971) and R. Stenberg (1998). It is motivated and studied for the Poisson equation in a polygonal domain with re-entrant corners. The method is extended to meshes being not quasi-uniform as well as to solutions with singularities occuring in presence of re-entrant corners. Finite elements on shape regular triangles and appropriate mesh grading near corners with singularities are used for the discretization process.
The properties of the discretization method are investigated and error estimates are derived. Numerical examples illustrating the method and the rates of convergence are given.

# On the computational cost of a local r-refinement method for variational problems

### Yves Tourigny

#### email: Y.Tourigny@bristol.ac.uk

Mesh relocation (r-refinement) methods have the reputation of being computationally costly compared to other adaptive approaches. This talk introduces a local r-refinement  method for variational problems and presents numerical examples in 2D. The assessment of the asymptotic computational complexity shows that for a certain class of problems, the proposed method beats global uniform h-refinement not only in accuracy but ultimately also in computational speed.

# A Parallel Domain Decomposition Preconditioner for Three-Dimensional Problems

### Peter Jimack

#### UK

We describe a parallel algorithm for the finite element solution of a general class of elliptic partial differential equations in three dimensions based upon a weakly overlapping domain decomposition preconditioner. The approach, which extends previous work for two-dimensional problems, is briefly described and numerical evidence is provided to demonstrate the potential of our proposed parallel algorithm.

Reference:

[1] Jimack, P K & Nadeem, S A: A Weakly Overlapping Parallel Domain Decomposition Preconditioner for the Finite Element Solution of Elliptic Problems in Three Dimensions. In Proceedings of the 2000 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA'2000), Volume III, ed. H.R. Arabnia (CSREA Press, USA), p.1517--1523, 2000. (http://www.comp.leeds.ac.uk/pkj/Papers/Conf-O/JN00.ps.gz)

# On H-Matrix FE Approximation to Nonlocal (Integral) Operators

### Boris N. Khoromskij

#### Germany

We discuss the H-matrix approximation of nonlocal (integral) operators on spatial domains or manifolds in Rd , d=2,3. The class of H-matrices allows a formatted matrix arithmetic of almost linear comlexity. In typical FEM and BEM applications the kernel of integral operator is defined by the fundamental solution (or by the Green's function) of the elliptic operator.

A systematic approach to derive data-sparse approximations of the matrix blocks in H-matrix techniques depending on the structural properties of the kernel function and the Galerkin ansatz space will be considered.
Among with the general low-rank matrix blocks based on the polynomial interpolation of the asymptotically smooth kernels, we then analyse the wire-basket as well as blended finite element and polynomial approximations. Applying the expansions of variable order, these techniques then lead to the accurate H-matrix approximation of almost linear complexity, O(N)-O(N log N),
for both the memory needs and the matrix-vector multiplication, where N is the problem size.

This talk represents joint work with Wolfgang Hackbusch, MPI Leipzig.

# Nonobtuse tetrahedral partitions  and the discrete maximum principle

### Michal Krizek

#### e-mail: krizek@math.cas.czkrizek@matsrv.math.cas.cz

We show how to generate refinements of tetrahedral partitions, where no obtuse angles appear. Such partitions play an important role in deriving discrete maximum principles and maximum norm error estimates for the finite element method.

We prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of the existence of some obtuse internal angle between faces of tetrahedra of triangulation of a given space domain.  This result represents a weakended form of the acute type contion for the three-dimensional case.

# Mesh generation and error estimation on anisotropic meshes

### Gerd Kunert

#### email: gerd.kunert@mathematik.tu-chemnitz.de

Adaptive algorithms are now widespread in finite element computation. However, for anisotropic solutions the design and analysis of robust methods is still at the beginning. Here we try to investigate two main ingredients together.

First we present the so--called Hessian strategy which has been used for some time to construct anisotropic meshes. We are particularly interested in the mesh information (i.e.~stretching direction, aspect ratio and element size).

The second part of the talk addresses several error estimators that are suitable for anisotropic finite element meshes. Some attention is given to the fact that, apparently, the anisotropic function and the anisotropic mesh have to correspond in a certain way. This correspondence is described by a so--called matching function.

Ultimately the error estimator should provide all information that are required for mesh refinement. Since this is not possible (yet?), the Hessian strategy is used instead to construct the new mesh. Our main result shows that such meshes are well aligned with the anisotropic solution, and reliable error estimation is possible.

References:

[1] Gerd Kunert: Anisotropic mesh construction and error estimation in the finite element method. Preprint SFB 393/00-01, TU Chemnitz, 2000. http://www-usercgi.tu-chemnitz.de/~gku/work/literatur.php3

[2] Gerd Kunert: A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. Logos Verlag, Berlin, 1999. http://archiv.tu-chemnitz.de/pub/1999/0012/index.html

### C. Schwab

#### email: schwab@sam.math.ethz.ch

We present a new algorithm for the efficient generation of the stiffness matrix in hp-FEM. This algorithm combines techniques from standard Galerin FEM (variable approximation order and variable numerical quadrature) with those from spectral methods (special shape functions and quadrature rules). For two- and three-dimensional problems, the stiffness matrix is generated in optimal complexity O(p^4) in 2D and  O(p^6)  in 3D, where p denotes the elemental polynomial degree.

References:

[1] J.M. Melenk, K. Gerdes, C. Schwab, Fully discrete hp-FEM: fast quadrature, in press in  Comput. Meths. Appl. Mech. Eng. (http://www.sam.math.ethz.ch/Reports/1999-15.html)

# Refined mixed finite element method for the Boussinesq equations in polygonal domains

### L. Paquet

#### F-59304 - Valenciennes Cedex, France

We consider the stationary equations of thermohydraulics in the setting of Boussinesq approximation with Dirichlet boundary conditions for the velocity and mixed Dirichlet and Neumann boundary conditions for the temperature. This problem and similar ones were recently studied where some existence and approximation results were obtained.

Here we consider the mixed formulation where the gradient of the velocity and the gradient of the temperature are introduced as new unknowns. In the mixed finite element method that we will consider to compute approximated solutions of the solutions, the velocity, the (kinematic) pressure, and the temperature are respectively approximated by piecewise constant fields, while gradients of temperature and velocity are approximated by Raviart-Thomas elements of degree 0.

Due to the corner points of the domain, the solutions of our problem are not regular in general. Nevertheless we show that they belong to appropriate weighted Sobolev spaces. These regularity results allow to formulate the precise mixed formulation of our problem. Next we derive appropriate refinement conditions on the regular family of triangulation in order to obtain quasi-optimal error estimates between a non-singular solution and the approximated one. A numerical test will be presented which confirms the theoretical convergence rates.

# The Residual Free Bubble Method for Bilinear Finite Elements

### Uwe Risch

#### email: uwe.risch@mathematik.uni-magdeburg.de

We analyze the Residual Free Bubble method (RFB method) applied to diffusion--convection equations with a dominating and skew-symmetric convection operator.
In RFB methods some standard FEM is modified by an enhancement of the discrete spaces in such a way that the differential equation will be satisfied exactly within each element.
The properties of the resulting schemes are well-known for P_1 elements (equivalence to a streamline diffusion approach, e.g. [4]) and to a certain extent for general simplicial elements[3].
In our talk, the RFB method is investigated especially in the case of Q_1 elements. The error analysis is based on a modification  (simplification) of the approach in [3]. The discussion of the stabilizing effect (not identical to a streamline diffusion one)  uses some unpublished results from [2].

References:

[1] U. Risch: The Residual Free Bubble Method for Bilinear Finite Elements. In preparation.
[2] L.P. Franca, L. Tobiska: Stability of the residual free bubble method for bilinear finite elements on rectangular grids. In preparation.
[3] F. Brezzi, D. Marini, E. S\"uli: Residual-free bubbles for advection-diffusion problems: The general error anlysis. Numer. Math. 85(2000), 31-47.
[4] F. Brezzi, T.J.R. Hughes, D. Marini, A. Russo, E. Süli: A priori error analysis of a finite element method with  residual-free bubbles for advection dominated equations. SIAM J. Numer. Anal. 36(1999), 1933-1984.

# Hybrid Coupled Domain Decomposition Methods

### Olaf Steinbach

#### email: steinbach@mathematik.uni-stuttgart.de

In this talk we discuss stability criteria for hybrid coupled domain decomposition methods, in particular for Mortar finite element methods. As it is well known, the BBL condition needed is equivalent to a stability estimate of a related projection operator in certain Sobolev spaces. We will present a new approach which is also applicable for nonuniform, i.e. adaptive meshes on the coupling boundaries. For locally quasiuniform meshes an additional  condition is needed which bounds the volume ratio of neighboured elements. We will discuss different choices of test and trial spaces for both primal and dual variables. For the local discretizaion of the subproblems one may use standard techniques such as finite or boundary element methods.

# Finite Element Methods and Ferrofluids

### Lutz Tobiska

#### email: tobiska@mathematik.uni-magdeburg.de

A basic model for the behaviour of magnetic fluids consisting of the Maxwell equations, the Navier-Stokes equations and the Young-Laplace equation on the free surface is described. If the temperature in the fluid is essential for the considered type of application, the model can also be extended  by the energy equation. Since ferrofluids are in general  nonconducting the Maxwell equations simplify to the equations for the magnetic potential with a nonlinear magnetization given by the Langevin function. Both the ferromagnetic and hydrodynamic properties of the magnetic fluid are essential for the understanding of the complex behaviour of these innovative fluids. The incompressible Navier-Stokes equations are appropriate extended to represent the influence of magnetic forces on the free surface of the fluids.
In particular we address to the following issues: discretization by stable finite elements, fast solutions by multigrid solvers and iterative methods for determing the free surfaces.

# Efficient and reliable approximations for elliptic eigenvalue problems

### Christian Wieners

#### email: christian.wieners@iwr.uni-heidelberg.de

We address two topics: in the first step, we consider efficient methods for approximating eigenfunctions of elliptic boundary value operators, and in the second step a method for the computation of a posteriori error bounds is proposed.

For the computation of the N  lowest eigenvalues and the corresponding eigenfunctions we use a nested inverse block iteration with a Ritz-Galerkin orthonormalization in every step; the arising linear problems are solved with a multigrid method. The eigenfunctions are approximated with P_2-elements.

On order to obtain error bounds, we use the method of complementary variational principles. Therefore, defect bounds are required, which are obtained by solving a dual problem with mixed RT_1-elements (using a hybrid multigrid method). Furthermore, a bound for the rest of the spectrum is required; this is obtain via domain homotopy.

Altogether, our method is of optimal complexity: we obtain guaranteed eigenvalue inclusions of accuracy O(1/n) with  O(n) operations.

The algorithm is implemented in the software system UG [1], which includes an efficient eigenvalue solver [2]. The second part of the talk is a joined work with M. Plum, H. Behnke, and U. Mertins [3].

References:

[1] P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuß,  H. Rentz-Reichert, C. Wieners:  UG -- a flexible software toolbox for solving partial differential  equations  Computing and Visualization in Science  1, (1997) 27-40

[2] P. Bastian, K. Johannsen, S. Lang, S. Nägele, V. Reichenberger, C.~Wieners, G.~Wittum, C. Wrobel: Advances in High-Performance Computing: Multigrid Methods for Partial Differential Equations and its Applications (2000) to appear in: High Performance Computing in Science and Engineering '00

[3] H. Behnke, U. Mertins, M. Plum, C. Wieners: Eigenvalue inclusions via domain decomposition (1999) to appear in Proc. of the Royal Soc. of London A

Last change: 07.08.2000,