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- ItemA logistic equation with nonlocal interactions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Caffarelli, Luis; Dipierro, Serena; Outrata, Jir̆íWe 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.
- ItemGraph properties for nonlocal minimal surfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Savin, Ovidiu; Valdinoci, EnricoIn 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.
- ItemError control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Müller, RüdigerA fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
- ItemStrong solutions for the interaction of a rigid body and a viscoelastic fluid(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Götze, KarolineWe 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.
- ItemDiffraction of stochastic point sets : exactly solvable examples(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Baake, Michael; Birkner, Matthias; Moody, Robert V.Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.
- ItemEnergy 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, TristanWe 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.
- ItemDiscretisation of the Maxwell equations on tetrahedral grids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2003) Schefter, JürgenThe aim of this report is to describe the discretisation of the Maxwell equations on tetrahedral grids with corresponding dual Voronoi cells to explain the resulting program. The symmetry of the coefficients of the matrix is proven. A small example shows an input file and same other details.
- ItemA 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, RüdigerUnderstanding 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.
- ItemHeuristic parameter selection based on functional minimization : optimality and model function approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Lu, Shuai; Mathé, KarstenWe analyze some parameter choice strategies in regularization of inverse problems, in particular the (modified) L-curve method and a variant of the Hanke-Raus rule. These are heuristic rules, free of the noise level, and they are based on minimization of some functional. We analyze these functionals, and we prove some optimality results under general smoothness conditions. We also devise some numerical approach for finding the minimizers, which uses model functions. Numerical experiments indicate that this is an efficient numerical procedure.
- ItemOptimal sensor placement: A robust approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Mohammadi, Masoumeh; Kanitsar, MartinWe 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.