Researchers from LNNC Stephane Lanteri and Frederic Valentin organized a mini-symposium at the International Conference on Spectral and High Order Methods (ICOSAHOM 2016) on “Hybridized and Multiscale Methods for Waves”.
The main objective of this minis-ymposium was to present some recent trends and contributions on hybridized methods for the numerical resolution of wave propagation PDE models. Different types of physical models were considered pertaining to time-domain or frequency-domain acoustics, elastodynamics and electromagnetics. The emphasis was put on methods able to deal accurately and efficiently with propagation problems involving complex geometrical features and heterogeneous media (unstructured mesh based methods) and/or high frequency fields (high order methods) and/or multiscale features (upscaling methods).
Two talks were also presented by LNCC researchers:
- “A MHM method for the Helmholtz equation”: the MHM method is a consequence of a hybridization procedure which characterizes the unknown as a direct sum of a global \coarse" solution and the solutions to local problems driven by the multipliers. As a result, the MHM method becomes a strategy that naturally incorporates multiple scales while providing solutions with high-order precision. s multiple scales while providing solutions with high-order precision. The completely independent local problems are embedded in the upscaling procedure. As such, the MHM method is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale boundary value problems with precision on coarse meshes. We highlight how the MHM method can be derived and analyzed within an abstract setting, and apply such a framework to devise a new MHM method for the Helmholtz equation. In the process, we recover some well-established and recent nite element methods. We assess theoretical results showing a large variety of numerical results for academic and highly heterogeneous coefficient problems.
- “A multiscale hybrid-mixed method for the Maxwell equations in time domain”: this work proposes a Multiscale Hybrid-Mixed (MHM) method for the Maxwell equation in time domain. The MHM method is a consequence of a hybridization procedure, and emerges as a method that naturally incorporates multiple scales while provides solutions with high-order precision. The computation of local problems is embedded in the upscaling procedure, which are completely independent and thus may be naturally obtained using parallel computation facilities. In this talk, we present the new MHM method for the two-dimensional Maxwell equations in time domain (Transverse Magnetic mode). We address some theoretical aspects of the method and propose an extensive numerical validation. We conclude that the MHM method is naturally shaped to be used in parallel computing environments and appears to be a highly competitive option to handle realistic multiscale hyperbolic boundary value problems with precision on coarse meshes.
ICOSAHOM 2016 took place in Rio de Janeiro (Brazil) from 27 June to 1 July.