The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems.

The omnipresence of lithium-ion batteries in mobile electronics, and hybrid and electric vehicles necessitates discovery of new lithium resources to meet rising demand and to diversify the global lithium supply chain. Here we demonstrate that lake sediments preserved within intracontinental rhyolitic calderas formed on eruption and weathering of lithium-enriched magmas have the potential to host large lithium clay deposits. We compare lithium concentrations of magmas formed in a variety of tectonic settings using in situ trace-element measurements of quartz-hosted melt inclusions to demonstrate that moderate to extreme lithium enrichment occurs in magmas that incorporate felsic continental crust. Cenozoic calderas in western North America and in other intracontinental settings that generated such magmas are promising new targets for lithium exploration because lithium leached from the eruptive products by meteoric and hydrothermal fluids becomes concentrated in clays within caldera lake sediments to potentially economically extractable levels.

Abstract. This paper outlines an energy-minimization finite-element approach to the modeling of equilibrium configurations for nematic liquid crystals in the presence of internal and external electric fields. The method targets minimization of system free energy based on the electrically and flexoelectrically augmented Frank-Oseen free energy models. The Hessian, resulting from the linearization of the first-order optimality conditions, is shown to be invertible for both models when discretized by a mixed finite-element method under certain assumptions. This implies that the intermediate discrete linearizations are well-posed. A coupled multigrid solver with Vanka-type relaxation is proposed and numerically vetted for approximation of the solution to the linear systems arising in the linearizations. Two electric model numerical experiments are performed with the proposed iterative solver. The first compares the algorithm's solution of a classical Freedericksz transition problem to the known analytical solution and demonstrates the convergence of the algorithm to the true solution. The second experiment targets a problem with more complicated boundary conditions, simulating a nano-patterned surface. In addition, numerical simulations incorporating these nano-patterned boundaries for a flexoelectric model are run with the iterative solver. These simulations verify expected physical behavior predicted by a perturbation model. The algorithm accurately handles heterogeneous coefficients and efficiently resolves configurations resulting from classical and complicated boundary conditions relevant in ongoing research.

SUMMARYThe incompressible. Stokes equations are a widely used model of viscous or tightly confined flow in which convection effects are negligible. In order to strongly enforce the conservation of mass at the element scale, special discretization techniques must be employed. In this paper, we consider a discontinuous Galerkin approximation in which the velocity field is H.div; /-conforming and divergence-free, based on the Brezzi, Douglas, and Marini finite-element space, with complementary space (P 0 ) for the pressure. Because of the saddle-point structure and the nature of the resulting variational formulation, the linear systems can be difficult to solve. Therefore, specialized preconditioning strategies are required in order to efficiently solve these systems. We compare the effectiveness of two families of preconditioners for saddle-point systems when applied to the resulting matrix problem. Specifically, we consider block-factorization techniques, in which the velocity block is preconditioned using geometric multigrid, as well as fully coupled monolithic multigrid methods. We present parameter study data and a serial timing comparison, and we show that a monolithic multigrid preconditioner using Braess-Sarazin style relaxation provides the fastest time to solution for the test problem considered.

To establish causality between flood basalt eruptions and extinction events and global environmental effects recorded by isotopic excursions in marine sediments, highly accurate and precise ages for the flood basalts are required. But flood basalts are intrinsically difficult to date. We illustrate how 40 Ar/ 39 Ar feldspar ages for silicic tuffs intercalated with and overlying sections of Steens Basalt, the earliest lavas of the Middle Miocene Columbia River Basalt Group in the northwestern United States, provide high-precision ages that, for the first time, make it possible to resolve age differences with stratigraphic position within a section of these flood lavas. The stratigraphically lowest rhyolitic tuff, a fall deposit, yielded an age of 16.592±±0.028 Ma (FCs = 28.02), and the uppermost, the alkali rhyolite ignimbrite Tuff of Oregon Canyon, is 16.468±±0.014 Ma. The argon and stratigraphic data indicate that Steens Basalt eruptions occurred from ~16.64 to 16.43 Ma in the southern end of its distribution. We estimate that the Steens Mountain geomagnetic reversal occurred at 16.496±±0.028 Ma (±0.18 Ma total error). Our estimates of the timing for initiation of volcanism and volumetric eruptive rates do not seem