A Hierarchical Multiscale Method for Nonlocal Fine-scale Models via Merging Weak Galerkin and VMS Frameworks

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Many problems in the natural sciences and engineering involve phenomena that possess a spectrum of material, spatial, and temporal scales which correspond to coarse and fine scale physics. Very often fine scale physics is also nonlocal and therefore these problems pose a great challenge to the current computational techniques. Nonlocality of fine-scales and advecting sharp gradients are two key ingredients in the modeling of several classes of fluid dynamics problems. This research effort focuses on the development of variationally based methods for problems with steep gradients and discontinuities in the underlying fields and wherein fine scales have a nonlocal feature. Of specific interest are propagating steep fronts for which interacting discontinuities challenge the stability of the numerical methods. Typical examples are chemically reacting fluids permeating through porous elastic solids where fast reaction rates produce steep concentration fronts. Such problems arise in (i) high temperature injection molding of fibrous composites in micro and nanomaterials engineering, and (ii) enhanced oil recovery and secondary shale gas recovery processes in petroleum engineering. Another class of problems from mathematical physics is advection dominated viscous flows leading to anisotropic turbulence. The PI will develop a unique and novel, simultaneous top-down and bottom-up multiscale approach for consistent representation of both the hierarchy of scales as well as the structure of inter-scale coupling operators for complex fluid mechanics problems. It blends ideas from the Variational Multiscale (VMS) method that helps decompose the governing system of equations into coarse-scale and fine-scale sub-problems and then employs Weak Galerkin (WG) ideas at the fine-scale variational level to extract models for the finer physics. Weak continuity of functions that is facilitated by the Weak Galerkin method results in fine-scale models that are nonlocal. A further generalization of WG via Discontinuous Galerkin (DG) ideas provides a framework to develop methods for problems with sharp gradients on sound mathematical basis. This notion leads to variationally derived Discontinuity Capturing (DC) methods that are independent of the user-defined or user-designed parameters. Emphasis is placed throughout on variationally consistent interscale coupling with rigorous treatment of the continuity conditions that are critical for the mathematical and algorithmic stability. The resulting computational algorithms will be ideal for massively parallel computing on distributed systems where message passing traditionally has been a bottle neck. The variational structures underlying the new methods will increase local-solves that are cost effective because of local resident memory on the new generation of processors while substantially reducing global communication between processors, thereby leading to efficient and economic computations. The mathematical frameworks and computational algorithms emanating from this work will be broadly disseminated by publication in high-quality archival journals, and by presentations at high impact conferences.