Abstract of POMS Project

The aim of this project is the implementation of an optimal Geometric Multigrid solver for large systems arising from B­Splines Galerkin approximation of Poisson's equation, which is the basis toward parallel and optimal fast solvers of discrete H1, H(curl) and H(div)­elliptic variationalproblems [Mazza].

High order Finite Elements methods, specially on (block) structured grids, are a good compromise to meet both high resolution (spectral convergence) and optimal scalability on the massively parallel hardware architecture of modern and future supercomputers. However, it is a known fact: matrices arising from High Order discretizations tend to have some pathologies in high frequencies. Such pathologies cannot be treated by the Multigrid method, which only deals with low frequencies.

In this work, we propose a solution to overcome these limitations using the GLT theory. The proposed algorithms were implemented in the CLAPP framework, developed within the NMPP and in collaboration with INRIA Sophia­Antipolis, INRIA Nancy Grand­Est and CEA­Cadarache. These algorithms are based on the Kronecker algebra and allow to handle complex geometries too. Therefore, they are suited for massive parallel computations. Moreover, the proposed restriction and prolongation operators allow for local refinement. Hence, there is full control on how the transfer is done which may be of great benefit for turbulence codes. The goal of this work is to port these developments using MPI+OpenMP. The resulting code will be shared and can be used in different production codes.

[Mazza]: M. Mazza, A. Ratnani, S. Serra­Capizzano. Spectral analysis and spectral symbol for the 2D curl­curl operator with application to the related iterative solutions. In progress