Vertebrate limb bud development - Computational advances and challenges in simulating organogenesis.
Dienstag, 3.5.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Limb bud development is regarded as an ideal model system to gain\ninsight into vertebrate organogenesis. Due to its simplicity, the limb\nbud has attracted theoretical modellers already decades ago and now all the\nmore in the advent of systems biology. Numerous genetic studies provide the\nbasic logic of signaling interactions as well as the domains of gene\nexpression [1, 2]. This has enabled us to build and experimentally test a\nreaction-diffusion PDE model of the molecular regulatory network that\ncontrols the initiation, propagation, and termination of signaling as well\nas the patterning during digit formation [3]. We are solving our models on\ngrowing domains of realistic shape using finite element methods. Given the\nsharp domain boundaries, travelling wave character of some solutions, and\nthe stiffness of the reactions we are facing numerous numerical challenges\nthat we will discuss.\n\n\n\n1. Rolf Zeller et al., Vertebrate Limb bud development: moving towards\nintegrative analysis of organogenesis. Nature Reviews Genetics 10, 845\n(2009).\n\n\n2. Jean-Denis Benazet et al., A self-regulatory system of interlinked\nsignaling feedback loops controls mouse limb patterning. Science 323, 1050\n(2009).\n\n\n3. Probst et al. Development, in press.\n
Scalable adaptive mesh refinement and large-scale applications in computational geosciences.
Dienstag, 17.5.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Many geophysical systems can be modeled by partial differential equations\n(PDEs) derived from fluid or solid mechanics. These systems give rise to\ncomplex multiscale behavior, which motivates to use of adaptive mesh refinement\nand coarsening (AMR) techniques. AMR refers to the ability to adapt the mesh\nresolution to local characteristics of the physical system, investing a dense\nmesh only in areas where high resolution is required, which can reduce the\nproblem size by several orders of magnitude. AMR is thus an attractive option\nfor the adequate discretization of localized multiscale phenomena, and even\nmore so if it supports dynamically changing the mesh to track moving features\nof the solution.\n\n\n\nThe challenge, however, lies in the fact that each processor in a parallel\nsimulation can only store a small part of the adaptive mesh, and that the\ntopological relations between mesh elements are irregular both within one and\nbetween neighboring processors. These facts entail complex storage and\ncommunication patterns, which have traditionally incurred a large computational\noverhead and severely limited the successful use of parallel dynamic AMR. I\nhave addressed these challenges by developing a collection of new parallel\ntechniques for forest-of-octree AMR, and I will describe the central ideas and\nessential concepts in this talk. I will conclude with a presentation of\nselected applications in mantle dynamics and seismic wave propagation.\n
Multi-level Monte Carlo methods for quantifying uncertainty in systems of conservation laws.
Dienstag, 21.6.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Central-Upwind Schemes for Shallow Water Models.
Dienstag, 12.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will first give a brief review on simple and robust central-upwind\nschemes for hyperbolic conservation laws.\n\nI will then discuss their application to the Saint-Venant system of\nshallow water equations. This can be done in a straightforward manner, but\nthen the resulting scheme may suffer from the lack of balance between the\nfluxes and (possibly singular) geometric source term, which may lead to a\nso-called numerical storm, and from appearance of negative values of the\nwater height, which may destroy the entire computed solution. To\ncircumvent these difficulties, we have developed a special technique,\nwhich guarantees that the designed second-order central-upwind scheme is\nboth well-balanced and positivity preserving.\n\nFinally, I will show how the scheme can be extended to the two-layer\nshallow water equations and to the Savage-Hutter type model of submarine\nlandslides and generated tsunami waves, which, in addition to the\ngeometric source term, contain nonconservative interlayer exchange terms.\nIt is well-known that such terms, which arise in many different multiphase\nmodels, are extremely sensitive to a particular choice their numerical\ndiscretization. To circumvent this difficulty, we rewrite the studied\nsystems in a different way so that the nonconservative terms are\nmultiplied by a quantity, which is, in all practically meaningful cases,\nvery small. We then apply the central-upwind scheme to the rewritten\nsystem and demonstrate robustness and superb performance of the proposed\nmethod on a number numerical examples.
Hybrid DG Methods for Incompressible Flow.
Dienstag, 26.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We propose a class of hybrid discontinuous Galerkin methods for the numerical\n solution of incompressible flow problems. Our approach yields consistent and locally\n conservative discretizations, and allows to treat non-conforming h and p-refinements\n in a natural way.\n\nBy explicit construction of a Fortin-operator, we show that the proposed methods\n are inf-sup stable with a constant which is independent of the meshsize, and only\n slightly dependent on the polynomial degree. This allows to derive a-priori error\n estimates for the Stokes problem, which are optimal in the mesh size and only\n slightly sub-optimal in the polynomial degree.\n\nWe also discuss the extension to stationary Oseen and Navier-Stokes equations, and illustrate our theoretical results with numerical tests.\n\n
Zweite Variation des Willmore-Funktionals
Dienstag, 2.8.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10