Proyecto Fondecyt regular “New Developments in Multiscale Hybrid-Mixed Finite Element Methods”
Solutions to physical models which address problems arising in industrial and/or natural settings often present strong variations in portions of space or time whose sizes vary by several orders of magnitude. Such problems are called multiscale problems. It is well known that classical numerical methods can produce poor approximations of solutions to multiscale problems when the influence of the physics at scales of smaller magnitude than the discretization scheme are improperly incorporated. In a broad sense, very fine meshes must be adopted to approximate the solution of multiscale problems.
The computational cost involved in multiscale simulations can be prohibitive when there is an imbalance between the computational architecture and and algorithm’s specific requirements for accuracy. Computational speed using a single processor is limited by technical issues such as heating. For this reason, there has been an extensive development of massively parallel computer architectures. Here, computers are built to leverage a large number of processors (grouped in cores) of mild speed and storage capacities. This new paradigm has led to a revision of what is expected from simulators from the viewpoint of numerical algorithms. Precision and robustness remain fundamental properties of numerical methods for extreme-scale computational science. However, the underlying algorithms must be naturally shaped to be distributed efficiently within these new massively parallel architectures.
Numerical methods built on the “divide-and-conquer” philosophy satisfy the architectural imperatives of the new generation of high-performance computers better than classical methods operating only on the finest scale of the discretization. Indeed, splitting the computation of extreme simulations into a set of independent problems of smaller size is a way to circumvent faults and to allow spatial and time data locality while taking full advantage (in terms of performance) of the granularity of the new generation of computer architectures.
In this context, the Multiscale Hybrid-Mixed (MHM) finite element method appear as an attractive “divide-and-conquer” option to handle multiscale problems by naturally merging the effects of multiple scales to provide solutions with high-order precision on coarse meshes. The underlying upscaling procedure transfers to a set of basis functions the responsibility of achieving high orders of accuracy at scales smaller than the coarse mesh. The upscaling is built inside the general framework of hybridization, in which the continuity of the solution is relaxed a priori and imposed weakly through the action of Lagrange multipliers. This characterizes the unknowns as the solutions of local problems with boundary conditions driven by the multipliers. The computation of local problems is embedded in the upscaling procedure, with local computations being completely independent and thus fitting naturally with parallel computation facilities.
In 2013 we presented the first version of the MHM method for the Laplace equation with highly oscillatory permiability coefficient. Since our first work the MHM method was extend for several problems (diffusion-advection-reaction problems, elasticity problem, Stokes problem, among others). However, the development of MHM methods for wave propagation and fluid flow models is not complete. Novel approaches to treat the time-dependence have not been explored and an effective extension of the MHM method to non-linear problems is not yet proposed.
The goal of the proposed project is to extend the scope of MHM methods to include nonlinear problems as well as new approaches to time variables. Mathematical and computational analyses will indicate the expected performance of these methods. In particular, we will develop and analyze novel multiscale finite element methods to approximate the solution of models describing seismic wave propagation in geological media, electromagnetic fields, and magneto-hydrodynamic models.
Our work can be divided into three main topics:
1. Derive, test, analyze, and obtain new theoretical results on MHM for wave propagation models and fluid flow models.
2. Define efficient algorithmic approaches for the computational implementation of the proposed numerical schemes (which include a novel linearization process, a novel time-domain discretization and a novel space adaptivity process).
3. Obtain useful simulations by implementing the proposed methods on a High-Performance Computing (HPC) architecture.
Nombre del proyecto: New Developments in Multiscale Hybrid-Mixed Finite Element Methods
Fuente de financiamiento: Conicyt
Investigador: Diego Paredes
Rol: Investigador Responsable
Fecha de inicio del proyecto: 02/04/2018
Fecha fin del proyecto: 31/03/2022