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Tales from the Dawn of Harrychical Finite Elements
Preconditioning Techniques Based on Auxiliary Discretizations (Abstract)
Infrastructures for Research Communities
Analytical and Geometrical Properties of Inverse Mean Curvature Flow (Abstract)
Variational Approximations in Quantum Dynamics
Maximum Norm Estimates for SVD Truncated Tensors (Abstract)
Striving for Simplicity: A Numerical Analyst Poaching in Mathematical Physics (Abstract)
reception in the mathematical library|
Room MA 163 (math building)
On the H1-stability of the L2-Projection onto Finite Element Spaces (Abstract)
Neugrüns Köche 
Schönhauser Allee 135 a, 10437 Berlin
Bus transfer from TU Berlin at 18.30
Randy Bank, Harry
On the H1-stability of the L2-Projection onto Finite Element Spaces
We study the stability in the $H^1$-seminorm of the $L_2$-projection onto finite element spaces, in the case of nonuniform but shape regular meshes in two and three dimensions. We prove stability for piecewise linear elements in 2d and 3d, and piecewise quadratic elements in 2d for meshes where neighboring elements differ by at most a factor of two in diameter. For less strongly graded meshes, our proof shows stability for a much broader class of finite element spaces.
Striving for Simplicity: A Numerical Analyst Poaching in Mathematical Physics
The numerical evaluation of operator determinants, originally born out of an attempt to validate some PDE calculations, has become a popular tool in areas of Mathematical and Theoretical Physics dealing with integrable systems; e..g., it was recently used by Nishigaki to calculate the `pion decay constant` of certain QCD-like theories. We review some of this development, tell an amusing story about the numerical evaluation of higher-order derivatives and its relation to a problem in graph theory, and show how a numerical analyst can be trapped into an attempt to simplify some purely mathematical theory.
Maximum Norm Estimates for SVD Truncated Tensors
The standard HOSVD truncation of tensors yields a controlled error with respect to the corresponding Hilbert norm, usually L2-norm. If the tensor represents a multivariate function, the aim may be to evaluate the function at certain points. Therefore, we need maximum error estimates instead. Usually, the maximum norm cannot be estimated by the L2-norm. We show that, nevertheless, this is possible for the HOSVD truncated tensors.
Analytical and Geometrical Properties of Inverse Mean Curvature Flow
When a hypersurface moves in normal direction with speed equal to the inverse of its (positive) mean curvature, the area of the surface grows exponentially everywhere while this nonlinear parabolic expansion is smoothing out the surface. The lecture gives an overview of properties of this flow including recent applications in Geometry and General Relativity.
Preconditioning Techniques Based on Auxiliary Discretizations
In this talk, I will report several recent results on using an auxiliary discretization as a main ingredient in designing preconditioner for discretized PDEs. One such result is that the discretized system for a class of 4th order problems can be preconditioned by using a mixed finite element discretization that is not always convergent as a discretization scheme alone. The general framework of Fast Auxiliary Space Preconditioning (FASP) methods will be used in the design and analysis of this type of methods.