
Chemnitz FEM Symposium 2000[ Announcement  Programme  List of participants ]Collection of abstracts 
The stress distribution at the top of a polyhedral corner or at a crack tip has the typical r^{alpha}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.
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.
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 NavierStokes equations with the finite element method
for the discretization of viscous terms. We use the nonconforming Crouzeix
 Raviart finite elements combined with the socallsed 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.
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 overshoots 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 residualbased 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+1dimensional numerical test cases.
References:
[1] W. Bangerth and G. Kanschat. Concepts for objectoriented finite element software  the deal.II library. Preprint 9943, SFB 359, Universität Heidelberg, Oct. 1999.
[2] R. Becker and R. Rannacher. A FeedBack Approach to Error Control in Finite Element Methods: Basic Analysis and Examples, EastWest J. Num. Math., 4:237264, 1996.
[3] R. Hartmann. Adaptive FEmethods 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 , 5:367386, 1995.
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 reentrant
corners. The method is extended to meshes being not quasiuniform as well
as to solutions with singularities occuring in presence of reentrant 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.
Mesh relocation (rrefinement) methods have the reputation of being
computationally costly compared to other adaptive approaches. This talk
introduces a local rrefinement 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 hrefinement not only in accuracy but ultimately also
in computational speed.
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 twodimensional 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.15171523, 2000. (http://www.comp.leeds.ac.uk/pkj/Papers/ConfO/JN00.ps.gz)
We discuss the Hmatrix approximation
of nonlocal (integral) operators on spatial domains or manifolds in R^{d}
, d=2,3. The class of Hmatrices 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 datasparse approximations of the matrix
blocks in Hmatrix techniques depending on the structural
properties of the kernel function and the Galerkin ansatz space will be
considered.
Among with the general lowrank matrix blocks based on the polynomial
interpolation of the asymptotically smooth kernels, we then analyse the
wirebasket as well as blended finite element and polynomial approximations.
Applying the expansions of variable order, these techniques then lead to
the accurate Hmatrix approximation of almost linear
complexity, O(N)O(N log N),
for both the memory needs and the matrixvector multiplication, where
N is the problem size.
This talk represents joint work with Wolfgang Hackbusch, MPI Leipzig.
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 threedimensional case.
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 socalled 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 socalled 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 393/0001, TU Chemnitz, 2000. http://wwwusercgi.tuchemnitz.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.tuchemnitz.de/pub/1999/0012/index.html
We present a new algorithm for the efficient generation of the stiffness
matrix in hpFEM. 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 threedimensional 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 hpFEM: fast quadrature, in press in Comput. Meths. Appl. Mech. Eng. (http://www.sam.math.ethz.ch/Reports/199915.html)
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 RaviartThomas 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 quasioptimal error estimates between a nonsingular solution and the approximated one. A numerical test will be presented which confirms the theoretical convergence rates.
We analyze the Residual Free Bubble method (RFB method) applied to diffusionconvection
equations with a dominating and skewsymmetric 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 wellknown 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: Residualfree bubbles for advectiondiffusion
problems: The general error anlysis. Numer. Math. 85(2000), 3147.
[4] F. Brezzi, T.J.R. Hughes, D. Marini, A. Russo, E. Süli: A
priori error analysis of a finite element method with residualfree
bubbles for advection dominated equations. SIAM J. Numer. Anal. 36(1999),
19331984.
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.
A basic model for the behaviour of magnetic fluids consisting of the Maxwell
equations, the NavierStokes equations and the YoungLaplace 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 NavierStokes 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.
For the computation of the N lowest eigenvalues and the corresponding eigenfunctions we use a nested inverse block iteration with a RitzGalerkin orthonormalization in every step; the arising linear problems are solved with a multigrid method. The eigenfunctions are approximated with P_2elements.
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_1elements (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. RentzReichert, C. Wieners: UG  a flexible software toolbox for solving partial differential equations Computing and Visualization in Science 1, (1997) 2740
[2] P. Bastian, K. Johannsen, S. Lang, S. Nägele, V. Reichenberger, C.~Wieners, G.~Wittum, C. Wrobel: Advances in HighPerformance 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