Discontinuous Galerkin methods on arbitrarily shaped elements and their application to interface problems.
Dienstag, 3.11.20, 14:15-15:15, Raum 226, virtuell Euwe
Motivated by the problem of numerical treatment of curved boundaries and interfaces in numerical PDEs, will review some recent work on the development of discontinuous Galerkin (dG) methods which are able to be applied on meshes comprising of essentially arbitrarily-shaped elements [1,2]. The use of such element shapes makes possible the fitted representation of curved geometries, by moving the variational crime challenge from the domain representation (as is the case for classical FEM/dG) to the quadrature evaluations. The second part of my talk will focus on the application of these ideas to the specific problem of proof of a posteriori error bounds for elliptic and parabolic interface problems on curved interfaces [3,4,5]. \n\n[1] A. Cangiani, Z. Dong, and E. H. Georgoulis. hp–Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Submitted for publication. PDF\n\n[2] A. Cangiani, Z. Dong, E. H. Georgoulis and P. Houston. hp–Version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathematics (2017)\n\n[3] A. Cangiani, E. H. Georgoulis, and Y. Sabawi. Adaptive discontinuous Galerkin methods for elliptic interface problems. Mathematics of Computation 87(314) pp. 2675 – 2707 (2018) \n\n[4] Stephen A. Metcalfe. Adaptive discontinuous Galerkin methods for nonlinear parabolic problems. PhD Thesis, University of Leicester (2015).\n\n[5] Younis A. Sabawi. Adaptive discontinuous Galerkin methods for interface problems. PhD Thesis, University of Leicester (2017).
Derivation of a bending plate model for nematic liquid-crystal elastomers via Gamma-convergence
Dienstag, 10.11.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract: Liquid-crystal elastomers (LCEs) are a class of materials, whose shape can be controlled via external stimulation. Here, we introduce a three-dimensional model describing the deformations. Its terms include the elastomer's hyperelastic energy (coupled to the liquid-crystal structure) and the liquid-crystal's Oseen-Frank energy. Using Gamma-convergence, we then derive and examine a dimension-reduced model, effectively describing the bending behaviour for thin LCE-plates.
Derivation of a bending plate model for nematic liquid-crystal elastomers via Gamma-convergence
Dienstag, 10.11.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract: Liquid-crystal elastomers (LCEs) are a class of materials, whose shape can be controlled via external stimulation. Here, we introduce a three-dimensional model describing the deformations. Its terms include the elastomer's hyperelastic energy (coupled to the liquid-crystal structure) and the liquid-crystal's Oseen-Frank energy. Using Gamma-convergence, we then derive and examine a dimension-reduced model, effectively describing the bending behaviour for thin LCE-plates.
Formoptimierung eindimensionaler Strukturen für den Haupteigenwert zweidimensionaler Gebiete
Dienstag, 17.11.20, 14:15-15:15, Hörsaal II, Virtual raum Lasker
Zusammenfassung: \nBetrachtet man die Eigenschwingungen einer dünnen Membran, so verändern sich diese durch die Anbringung einer eindimensionalen Versteifung. Aus diesem Sachverhalt lässt sich ein Formoptimierungsproblem für den Haupteigenwert der Eigenschwingungen herleiten. Während es noch recht einfach ist die Existenz geeigneter Lösungen nachzuweisen, so ist es umso schwerer Aussagen über die Strukturen optimaler eindimensionaler Mengen zu treffen. Wir werden in diesem Vortrag Eigenschaften der Löser des Formoptimierungsproblems analytisch untersuchen und durch numerische Experimente die Struktur möglicher Optimierer besser kennenlernen.\n
A bounded numerical solution with a small mesh size implies existence of a smooth solution to the Navier-Stokes equations
Dienstag, 1.12.20, 14:15-15:15, Hörsaal II, Virtual Raum Lasker
Abstract: We prove that for a given smooth initial value, if the finite element solution of the three-dimensional Navier-Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier-Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.
On variational models for martensitic inclusions
Dienstag, 8.12.20, 13:30-14:30, Hörsaal II (virtuell:Lasker)
Shape-memory alloys are special materials that are able to "remember" their original shapes after deformation. Microstructures in these materials are often modeled in the context of the calculus of variations by singularly perturbed multiwell elastic energy functionals. \nIn this talk, I shall discuss recent analytical results on variational models for martensitic nuclei based on linearized elasticity, and solutions to related partial differential inclusion problems.
Linear and nonlinear methods for model reduction
Dienstag, 15.12.20, 14:15-15:15, Hörsaal II (virtuell:Lasker)
We consider reduced order methods for the approximation of classes of high-dimensional functions, such as solutions of parametric PDEs. The usual approach to model reduction for parametric PDEs is to construct a low dimensional linear space Vn which accurately approximates the solution manifold and use it to built an effcient forward solver. In some cases, the construction of one suitable linear space Vn is not feasible numerically, for instance if the target accuracy is too small. It is well-known that nonlinear methods may provide improved effciency. In a so-called library approximation, the idea is to replace Vn by a collection of linear (or affine) spaces V1, . . . , VN of dimension m < n.\n\nIn this talk, we first introduce various analytic anisotropic model classes based on Taylor expansions and study their approximation by finite dimensional polynomial spaces PΛ described by lower sets Λ of cardinality n. Then, in the framework of parametric PDEs, we present a possible strategy that can be used to built a library and provide an\nanalysis of its performance.\n\nThis is a joint work with: A. Bonito, A. Cohen, R. DeVore, P.\nJantsch, and G. Petrova.
Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis
Dienstag, 12.1.21, 14:00-15:00, Hörsaal II (virtuell:Lasker)
Fueled by many applications in random processes, imaging science, geophysics, etc., fractional Laplacians have recently received significant attention. The key driving force behind the success of this operator is its ability to capture non-local effects while enforcing less smoothness on functions. In this paper, we introduce a spectral method to approximate this operator employing a sinc basis. Using our scheme, the evaluation of the operator and its application onto a vector has complexity of O(Nlog(N)) where N is the number of unknowns.\nThus, using iterative methods such as CG, we provide an efficient strategy to solve fractional partial differential equations with exterior Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works in both 2d and 3d. We also recover the FEM rates of convergence on benchmark problems. We further illustrate the efficiency of our approach by applying it to fractional Allen-Cahn and\nimage denoising problems.\n
Colour Image Denoising: Numerical Approximation of the Constrained Total Variation Flow
Dienstag, 26.1.21, 14:15-15:15, Hörsaal II (virtuell:Lasker)
The Rudin-Osher-Fatemi model is a well-known denoising model for greyscale images. One approach to adapt this model to colour images involves computing the total variation flow constrained to a manifold. In a recent work by Giga et al., an algorithm to approximate the constrained total variation flow is proposed and a discretisation using piecewise constant functions is presented. As piecewise constant functions are generally not suited to approximate functions of bounded variation, we present a similar algorithm using piecewise affine finite elements, in particular Crouzeix-Raviart elements. We show stability of the scheme, experimental convergence rates and some improved denoising results. This work is the result of my master thesis.
Singular Solutions and Adaptive Approximations of Total Variation Problems
Dienstag, 2.2.21, 14:15-15:15, Hörsaal II (virtuell:Lasker)
Total variation problems are often encountered in image denoising, for example in the well-known Rudin-Osher-Fatemi model. Since the model admits all functions of bounded variation, the P1 finite element method converges suboptimal in this case. In recent works from Chambolle & Pock and Bartels, the approximation via Crouzeix-Raviart finite elements was analyzed and error estimates were established if the dual solution of the problem is Lipschitz continuous.\n\nWe will show an example with a non-Lipschitz continuous dual solution of the Rudin-Osher-Fatemi model on which the optimal convergence rate is still achieved. With a graded grid approach, we are able to improve the converge rates from uniform grids on examples where the primal solution is piecewise constant. With a reconstruction formula for the Crouzeix-Raviart finite elements to obtain a feasible dual solution, we are able to use the dual-gap error estimator presented in a recent work from Bartels & Milicevic to adaptively refine the mesh grid. This more general approach results in improved experimental convergence rates, which are slightly slower than the obtained convergence rates on graded grids. This is the presentation of the results of my master thesis.\n
Dienstag, 9.2.21, 14:15-15:15, Hörsaal II (virtuell:Lasker)
Variational convergences for functionals and differential operators depending on vector fields
Dienstag, 9.2.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
In this seminar, I present an extract of my PhD thesis, which concerns variational convergences for functionals and differential operators depending on a family of locally Lipschitz continuous vector fields X. This setting was introduced by Folland and Stein and has recently found numerous applications in the literature. The convergences taken into account date back to the 70’s and are Γ-convergence, introduced by Ennio De Giorgi and Tullio Franzoni, dealing with functions and functionals, and H-convergence, whose theory was initiated by François Murat and Luc Tartar and which deals with differential operators.\nThe main result presented today, under a linear independence condition on the family of vector fields X, is a Γ-compactness theorem and ensures that sequences of integral functionals depending on vector fields, with standard regularity and growth conditions, Γ-converge in the strong topology of Lp, up to subsequences, to a functional belonging to the same class.\nAs an interesting application of the Γ-compactness theorem, I finally show that the class of linear differential operators in X-divergence form is closed in the topology of the H-convergence. The variational technique adopted to this aim relies on a new approach recently introduced by Nadia Ansini, Gianni Dal Maso and Caterina Ida Zeppieri.