Interface Problems for Dispersive equations

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck, Pelloni and Sheils (2014). The dispersive nature of the problem presents additional difficulties that are addressed here.Comment: 18 pages, 6 figures. arXiv admin note: text overlap with arXiv:1402.3007, Studies in Applied Mathematics 201

Fractal homogenization of multiscale interface problems

Inspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of non-separating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the solution of the asymptotic limit problem and exponential convergence of the approximating finite-scale solutions. Computational experiments involving a related numerical homogenization technique illustrate our theoretical findings

Hybridized CutFEM for Elliptic Interface Problems

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples

HPC compact quasi-Newton algorithm for interface problems

In this work we present a robust interface coupling algorithm called Compact Interface quasi-Newton (CIQN). It is designed for computationally intensive applications using an MPI multi-code partitioned scheme. The algorithm allows to reuse information from previous time steps, feature that has been previously proposed to accelerate convergence. Through algebraic manipulation, an efficient usage of the computational resources is achieved by: avoiding construction of dense matrices and reduce every multiplication to a matrix-vector product and reusing the computationally expensive loops. This leads to a compact version of the original quasi-Newton algorithm. Altogether with an efficient communication, in this paper we show an efficient scalability up to 4800 cores. Three examples with qualitatively different dynamics are shown to prove that the algorithm can efficiently deal with added mass instability and two-field coupled problems. We also show how reusing histories and filtering does not necessarily makes a more robust scheme and, finally, we prove the necessity of this HPC version of the algorithm. The novelty of this article lies in the HPC focused implementation of the algorithm, detailing how to fuse and combine the composing blocks to obtain an scalable MPI implementation. Such an implementation is mandatory in large scale cases, for which the contact surface cannot be stored in a single computational node, or the number of contact nodes is not negligible compared with the size of the domain. \c{opyright} Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Comment: 33 pages: 23 manuscript, 10 appendix. 16 figures: 4 manuscript, 12 appendix. 10 Tables: 3 manuscript, 7 appendi