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End date: 2019 × Start date: 2011 × DDC: 510 × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 10 of 1042
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    A logistic equation with nonlocal interactions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Caffarelli, Luis; Dipierro, Serena; Outrata, Jir̆í
    We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.
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    Graph properties for nonlocal minimal surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico
    In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.
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    Strong solutions for the interaction of a rigid body and a viscoelastic fluid
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Götze, Karoline
    We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.
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    Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Giesselmann, Jan; Pryer, Tristan
    We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of AllenCahn/CahnHilliard/NavierStokesKorteweg type which allows for phase transitions.We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.
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    A new perspective on the electron transfer: Recovering the Butler-Volmer equation in non-equilibrium thermodynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dreyer, Wolfgang; Guhlke, Clemens; Müller, Rüdiger
    Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions and adsorption but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.
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    Optimal sensor placement: A robust approach
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Mohammadi, Masoumeh; Kanitsar, Martin
    We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.
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    A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.
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    On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Linke, Alexander; Rebholz, Leo G.; Wilson, Nicholas E.
    It was recently proven that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be gamma^-frac12 (where gamma is the stabilization parameter), the computational results suggest the rate may be improvable gamma^-1. We prove herein the analytical rate is indeed gamma^-1, and extend the result to other incompressible flow problems including Leray-alpha and MHD. Numerical results are given that verify the theory.
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    Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, Jürgen
    In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.
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    Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Eigel, Martin; Grasedyck, Lars; Gruhlke, Robert; Moser, Dieter
    We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length.