Research Seminar SoSe 2014

Abstracts

Shuying Zhai: A block-centered finite difference method for time-fractional diffusion equation on nonuniform grids

A block-centered finite difference method is proposed for the time-fractional diffusion equation with α∈(0,1) and Neumann boundary conditions on nonuniform grids. The method is unconditionally stable and second-order accurate in space and (2-α)-order accurate in time respectively. Numerical experiments are carried out to support the theoretical analysis. Particularly, for the boundary layer and high gradient problems, it shows that numerical results obtained using nonuniform grids are significantly more accurate than those using uniform grids.

Georgi Georgiew: Separiertes Markieren für adaptive Finite Elemente Methoden

Motivation des Separierten Markierens, Definition des Separierten Markierens und des Thresholding Second Algorithm, Implementation und numerische Experimente, Modifikationen am Algorithmus, Fehlerschätzer in der H(div)-Norm und abschließende Betrachtungen.

Dietmar Gallistl: Adaptive nonconforming finite element approximation of eigenvalue clusters

his talk presents optimal convergence rates of an adaptive nonconforming FEM for eigenvalue clusters. New techniques from the medius analysis enable the proof of L² error estimates and best-approximation properties for nonconforming finite element methods and thereby lead to the proof of optimality. Applications include the nonconforming P₁ FEM for the eigenvalues of the Stokes system and the Morley FEM for the eigenvalues of the biharmonic operator. The optimality in terms of the concept of nonlinear approximation classes is concerned with the approximation of invariant subspaces spanned by eigenfunctions of an eigenvalue cluster. In order to obtain eigenvalue error estimates, this talk presents new estimates for nonconforming finite elements which relate the error of the eigenvalue approximation to the error of the approximation of the invariant subspace.

Georgi Mitsov: Nonlinear Curl-Curl Problems

The question of the existence of localized in space solutions of the Maxwell’s equations in nonlinear dispersive media, has engaged the interest of researchers both in mathematics and in physics. In a Kerr-type nonlinear medium, solving the Maxwell’s equations for a monochromatic fields with a real valued complex amplitudes, simplifies to solving the semilinear elliptic equation ∇×∇× E + V (x)E = Γ(x)|E|2E, (1) for a vector field E :ℝ³ → ℝ³ , where ∇× operator, V and Γ belong to denotes the curl L^∞(\ℝ³, ℝ). A unitarian field theory for classical electrodynamics, introduced by Benci and Fortunato in 2004, considers a semilinear perturbation of the Maxwell’s equations (SME). The nonlinearity yields the existence of a finite-energy solitary waves of the SME, by which the particles are described. In the magnetostatic case, the SME reduce to the semilinear elliptic equation ∇×∇× A = f (A), (2) here f' is the gradient of a smooth function f : ℝ → ℝ, and A :ℝ³ → ℝ³ is the gauge potential related to the magnetic field. Both problems imply specific conditions for their characterizing parameters, V and Γ and f , with regard to the physical situations they describe. However, from a more general mathematical perspective equations (1) and (2) share common difficulties. Existence of nontrivial finite-energy solutions of a slightly generalized version of equation (1) ∇×∇× u + V (x)u = Γ(x)|u|p−1u, is proven for three types of coefficients V and Γ, and an exponent p ∈ (1, 5). The approach is taken from a variational calculus perspective. Applying a special cylindrically-like symmetric ansatz, the initial vector-fields variational problem is restricted to a scalar-functions one. Further, the latter variational problem is reduced to a minimization problem on a constraint manifold. By using concentration compactness techniques and tools from the nonlinear functional analysis the existence of minimizers is attained. The reduced by the ansatz dimen sionality is suitable for a numerical treatment. As a starting point, the method of steepest descent for finding local minimizers is presented, as well as an outlook for more advanced and comprehensive investigations.

Karoline Köhler: Reliable and Efficient Error Control for the Obstacle Problem

This talk presents some results for the error analysis of the obstacle problem, which is the protoypical example of a variational inequality. The first part of the talk deals with a general reliable and efficient error estimate, independent of any specific finite element method. Therein the efficiency is understood in terms of the error u − v in H₀¹(Ω) for the exact solution u and an approximation v in H₀¹(Ω) in the primal variable and the error λ − μ for the exact Lagrange multiplier λ and an approximation μ in the dual space H¹(Ω). This leads to the equivalence |||u − v||| + |||λ − μ|||_∗ ≈ computable terms possibly up to multiplicative generic constants. This general result is applied to the conforming Courant finite element method (FEM) in the second part of the talk, where the efficiency is analysed beyond the aforementioned equivalence and leads to |||u − v||| ≈ computable terms. Numerical experiments which demonstrate and highlightthe results for various finite element methods conclude the talk.

Christian Kreuzer: Design and convergence analysis for an adaptive discretization of the heat equation

We present an adaptive fully discrete space-time finite element method for the heat equation. The algorithm is based on a classical adaptive time-stepping scheme supplemented by an additional control of a potential energy increase of the discrete solution originating from coarsening of the spatial meshes. This control allows to prove critical energy estimates in terms of given data from which one can derive an apriori computable minimal time-step-size, which is sufficient for the required tolerance. The minimal step-size is used by the algorithm and guarantees that the final time is reached in finitely many time-steps and within a prescribed tolerance.
The minimal time-step-size has also a very positive effect in simulations. We present numerical experiments that show a significant speedup compared to classical time-stepping schemes since too small time-steps are avoided.

Dietrich Braess: A posteriori Fehlerschätzer für discontinuous Galerkin Elemente mit dem Zwei Energien Prinzip

Es gibt mehrere Varianten von DG Elementen, die sich durch die Strafterme unterscheiden, sich aber nach Arnold u.a. einheitlich im Rahmen von gemischten Methoden behandeln lassen. Die a posteriori Schätzer nach dem Zwei-Energien Prinzip haben gegenüber den residualen Schätzern mehrere Vorteile.
1. Sie enthalten als obere Schranken keine generischen Konstanten.
2. Sie sind p-robust, d.h. die Effizienz ist unabhängig vom Polynomgrad.
3. Der Beweis für die Abschätzung noch oben ist einfacher als beim residualen.
4. Es gibt einen Vergleich zwischen DG Verfahren und konformen Lagrange-Elementen.

Dirk Pauly: On the Maxwell constants in 3D

We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincar\'e's constants which can be bounded from above by $\diam/\pi$. In other words, the second and positive Maxwell eigenvalue lies between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue. This new estimate has applications not only to analysis itself but also to numerical analysis like preconditioning or functional a posteriori error estimates for Maxwell's equations.