Diffusion in networks with time-dependent transmission conditions

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.Comment: corrected typos, references removed, revised Lemma A.3. Appl. Math. Optim. (2013

Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions

Optimized Schwarz Waveform Relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to a mathematically hard best approximation problem of homographic type. We study in this paper in detail this problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel type. We give for each case closed form asymptotic values for the parameters, which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, and we measure performance corresponding to our analysisComment: 42 page

On Discontinuous Dirac Operator with Eigenparameter Dependent Boundary and Two Transmission Conditions

In this paper, we consider a discontinuous Dirac operator with eigenparameter dependent both boundary and two transmission conditions. We introduce a suitable Hilbert space formulation and get some properties of eigenvalues and eigenfunctions. Then, we investigate Green's function, resolvent operator and some uniqueness theorems by using Weyl function and some spectral data

Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n 652, and a sequence (Kj) of compact sets converging to an (n-1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on \u3a9\Kj, with Neumann boundary conditions on 02(\u3a9\Kj). We prove that the limit of these solutions is a minimiser of the same functional on \u3a9\M subjected to a transmission condition on M, which can be expressed through a measure \ub5 supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure \ub5 and the sequence (Kj) is expressed by means of suitable local minimum problems