Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media

In this paper we propose and analyze a preconditioner for a system arising from a finite element approximation of second order elliptic problems describing processes in highly het- erogeneous media. Our approach uses the technique of multilevel methods and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus (see [8]). The main results are the design and a theoretical and numerical justification of an iterative method for such problems that is robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient (related to the permeability/conductivity).Comment: 28 page

On Parallel Numerical Algorithms for Fractional Diffusion Problems

Proceedings of: Third International Workshop on Sustainable Ultrascale Computing Systems (NESUS 2016). Sofia (Bulgaria), October, 6-7, 2016.In this work, we consider a parallel numerical solution of problems depending onf ractional power sof elliptic operator. Three different state of theart approaches are used to transform the original non-local problem into well-known local PDE problems. Parallel numerical algorithms for allt hreea pproaches are developed and discussed. Results of their parallel performance tests are presented and analysed.The work presented in this paper has been partially supported by EU under the COST programme Action IC1305,’Network for Sustainable Ultrascale Computing (NESUS)’. This research was also funded by agrant (No. MIP-074/2015) from the Research Council of Lithuania

Analysis of fiber-reinforced concrete: micromechanics, parameter identification, fast solvers

Proceedings of: Third International Workshop on Sustainable Ultrascale Computing Systems (NESUS 2016). Sofia (Bulgaria), October, 6-7, 2016.Ultrascale computing is required for many important applications in chemistry, computational fluid dynamics etc., see an overview in the paper Applications for Ultrascale Computing by M. Mihajlovic et al. published in the International Journal Supercomputing Frontiers and Innovations, Vol 2 (2015). In this abstract we shortly describe an application that involves many aspects described in the above paper - the multiscale material design problem. The problem of interest is analysis of the fiber reinforced concrete and we focus on modelling of stiffness through numerical homogenization and computing local material properties by inverse analysis. Both problems require a repeated solution of large-scale finite element problems up to 200 million degrees of freedom and therefore the importance of HPC and ultrascale computing is evident.The work is supported by COST Action IC1305 project Network for Sustainable Ultrascale Computing and a bilateral project of collaboration between the Institute of Geonics CAS and IICT BAS. Further support is through the projects LD15105 Ultrascale computing in geosciences and LQ1602 IT4Innovations excellence in science supported by the Ministry of Education, Youth and Sports of the Czech Republic