Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bound

Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'

Slides 18th CMCIM 2024, Toward an algebraic multigrid method for the indefinite Helmholtz equation

International audienceIt is well known that multigrid methods are very competitive in solving a wide range of SPD problems. However achieving such performance for non-SPD matrices remains an open problem. In particular, three main issues may arise when solving a Helmholtz problem : some eigenvalues may be negative or even complex, requiring the choice of an adapted smoother for capturing them, and because the near-kernel space is oscillatory, the geometric smoothness assumption cannot be used to build efficient interpolation rules. Moreover, the coarse correction is not equivalent to a projection method since the indefinite matrix does not define a norm. We present some investigations about designing a method that converges in a constant number of iterations with respect to the wavenumber. The method builds on an ideal reduction-based framework and related theory for SPD matrices to improve an initial least squares minimization coarse selection operator formed from a set of smoothed random vectors. A new coarse correction is proposed to minimize the residual in an appropriate norm for indefinite problems. We also present numerical results at the end of the paper

Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue Numerical algorithms for high-performance computational science