Multigrid methods, algebraic or geometric, commonly suffer from high frequency residuals after prolongation. This paper develops a stable approach to remove high frequency residuals for geometric multigrid methods for solving nonlinear advection–diffusion problems with degenerate coefficients. Here, a local problem is treated by optimization on subdomains with mesh refinements. Newton's method is utilized in the procedure and the iteration is completed when the residual in the subdomain is reduced to the given magnitude, usually set to be the average of residuals in the non-high-frequency domains. An oversampling technique is employed to further improve the stability by providing a definite flow path in regions where coefficients have high contrast and complex structures. Removing high frequency residuals before continuing the global Newton iteration improves global convergence behavior.

Diabetic encephalopathy (DE) is a severe complication of diabetes mellitus affecting the central nervous system (CNS), characterized by cognitive dysfunction. This review systematically explores how the interplay between brain insulin resistance (BIR) and neuroinflammation contributes to the pathogenesis of DE. BIR refers to the diminished responsiveness of the CNS to insulin signaling, resulting in the suppression of the PI3K/Akt and MAPK pathways, impaired glucose metabolism, and dysfunction across multiple neural cell types—including neurons, astrocytes, microglia, brain endothelial cells, and oligodendrocytes. These disturbances manifest as impaired energy metabolism, compromised synaptic plasticity, disruption of the blood–brain barrier, and reduced myelination. Importantly, BIR and neuroinflammation form a vicious cycle within these cells, mutually exacerbating each other and jointly driving the pathological progression of DE. Finally, we have compiled a list of currently available drugs that can improve BIR and suppress neuroinflammation, along with the latest progress in clinical trials, to provide new insights for the future of precision treatment for DE.

Summary: In this paper, we propose a fully implicit space-time multiscale scheme to improve computational efficiency in solving nonlinear multiphase flow in porous media. Here, error estimators are used for adaptively changing the spatial-temporal mesh. This algorithm applied to the black-oil model is compared to a standard control volume approach using a fine time and spatial mesh. Error estimators are introduced to determine subdomains of the reservoir in which high nonlinearity hinders Newtonian convergence. This is followed by applying local fine timesteps to these marked regions, whereas the remaining regions retain the coarse time scale. Once a temporal discretization is determined for different parts of the reservoir, the spatial mesh is refined for treating saturation fronts. The nonmatching interfaces arising from different temporal and spatial scales are resolved by the enhanced velocity method, which enforces strongly the continuity of fluxes. This whole system is solved monolithically. Results from a three-phase black-oil model are described. The multiscale solution is compared to a uniformly fine spatial mesh and fine timestepping solution to confirm accuracy. Solutions from both Gaussian and channelized permeability fields are presented. In the multiscale solution, we observe temporal refinements being applied to the water and gas saturation fronts by reducing the timestep size to guarantee Newtonian convergence. We also observe that the extent of refinements is balanced between the two saturation fronts based on their respective nonlinearity. For Gaussian permeability fields, the algorithm only treats saturation fronts spatially, whereas for channelized permeability fields, the geological features are also considered. Production profiles for the three phases match well between the two solutions under specific refinement criteria. We also investigate the improvement in computational efficiency, as well as the algorithm scalability in regard to increasing problem sizes. We observe a speedup of approximately 10 for solving the linear system, and such speedup increases as the problem size expands. In reservoir simulation, schemes that attempt to decouple the diffusion and advection process become less robust as the physics becomes more complex. Our approach, which is focused on handling the high nonlinearities, can be treated fully implicitly and provide a reduction in simulation costs. This approach can also be applied to model order reduction in providing accurate snapshot solutions. NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.