# HOMAR Associate Team

## High performance Multiscale Algorithms for wave pRopagation problems

## Scientific context

The general scientific context of this collaboration is the study of time dependent wave propagation problems with strong multiscale features (in space and time). The general goal is the design, analysis and implementation of a family of innovative high performance numerical methods particularly well suited to the simulation of multiscale wave propagation problems. Mathematical models based on partial differential equations (PDE) embedding multiscale features occur in a wide range of scientific and technological applications involving wave propagation in heterogeneous media.

Finite element methods are frequently adopted to approximate the solution of such PDE models. Nevertheless, it is well-known that the accuracy of numerical solutions may seriously deteriorate on coarse meshes when they show multiscale or high-contrast features. Such an issue has led to the concept of multiscale basis functions in the seminal work [Babuska and Osborn, 1983] (further extended to the two-dimensional case in [Hou and Wu, 1997]) and allowed numerical methods to be precise on coarse meshes. Such physically rooted basis functions are the counterpart of polynomial basis functions generally adopted in classical finite element schemes, which are defined on a purely algebraic setting. Recently, researchers at LNCC introduced a family of finite element methods particularly adapted to be used in high-contrast or heterogeneous coefficients problems, [Araya et al., 2013]-[Harder et al., 2013], named Multiscale Hybrid-Mixed (MHM) methods. These novel finite element methods share the following properties: (i) Stable and high-order convergent; (ii) Accurate on coarse meshes; (iii) Naturally adapted to high-performance parallel computing; (iv) Induce a face-based a posteriori error estimator (to drive mesh adaptivity); (v) Locally conservative.

## Objectives

The main scientific goals of this collaboration are:

- To design and analyze new MHM methods for the system of time-domain Maxwell equations coupled to models of physical dispersion, in view of their application to light interaction with nanometer scale structures;
- To design and analyze new MHM methods for the system of time-domain elastodynamic equations for modeling elastic wave propagation in anisotropic media;
- To devise appropriate discrete versions of the proposed MHM methods using DG (Discontinuous Galerkin) formulations for the discretization of the local solvers, and to study the mathematical properties (stability, convergence) of the combined MHM-DGTD strategies;
- To implement the proposed MHM-DGTD solution strategy on modern parallel computing architectures combining coarse grain (MIMD - Multiple Instruction Multiple Data) and fine grain (SIMD - Single Instruction Multiple Data) processing units;
- To demonstrate the capabilities of the developed MHM-DGTD parallel solution strategies for the simulation of selected problems in the fields of nanophotonics and elastodynamics.

## Sponsors

HOMAR is funded by Inria's Direction of the European and International Partnerships Department and by FAPERJ (Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro).