Thomas R. Benson

MONOLITHIC MULTIGRID METHODS FOR TWO-DIMENSIONAL RESISTIVE MAGNETOHYDRODYNAMICS ∗

Magnetohydrodynamic (MHD) representations are used to 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. The resulting linear systems that arise from discretization and linearization of the nonlinear problem are generally difficult to solve. In this paper, we investigate multigrid preconditioners for this system. We consider two well-known multigrid relaxation methods for incompressible fluid dynamics: Braess--Sarazin relaxation and Vanka relaxation. We first extend these to the context of steady-state one-fluid viscoresistive MHD. Then we compare the two relaxation procedures within a multigrid-preconditioned GMRES method employed within Newtons method. To isolate the effects of the different relaxation methods, we use structured grids, inf-sup stable finite elements, and geometric interpolation. We present convergence and timing results for a t...

Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects

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 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 multigrid solver. The first compares the algorithms solution of a classical Freedericksz transition problem to the known analytical solution and demons...

Preconditioning a mass‐conserving discontinuous Galerkin discretization of the Stokes equations

Summary The 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,Ω)-conforming and divergence-free, based on the Brezzi, Douglas, and Marini finite-element space, with complementary space (P0) 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. Copyright

Secure distributed membership tests via secret sharing: How to hide your hostile hosts: Harnessing shamir secret sharing

Data security and availability for operational use are frequently seen as conflicting goals. Research on searchable encryption and homomorphic encryption are a start, but they typically build from encryption methods that, at best, provide protections based on problems assumed to be computationally hard. By contrast, data encoding methods such as secret sharing provide information-theoretic data protections. Archives that distribute data using secret sharing can provide data protections that are resilient to malicious insiders, compromised systems, and untrusted components. In this paper, we create the Serial Interpolation Filter, a method for storing and interacting with sets of data that are secured and distributed using secret sharing. We provide the ability to operate over set-oriented data distributed across multiple repositories without exposing the original data. Furthermore, we demonstrate the security of our method under various attacker models and provide protocol extensions to handle colluding attackers. The Serial Interpolation Filter provides information-theoretic protections from a single attacker and computationally hard protections from colluding attackers.