Eric C Cyr

A first-order system least-squares finite element method for the Poisson-Boltzmann equation

The Poisson‐Boltzmann equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson‐Boltzmann equation. We expose the flux directly through a first‐order system form of the equation. Using this formulation, we propose a system that yields a tractable least‐squares finite element formulation and establish theory to support this approach. The least‐squares finite element approximation naturally provides an a posteriori error estimator and we present numerical evidence in support of the method. The computational results highlight optimality in the case of adaptive mesh refinement for a variety of molecular configurations. In particular, we show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole, which highlights robustness of our approach.

A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD

The one-fluid visco-resistive MHD model provides a description of the dynamics of a charged fluid under the influence of an electromagnetic field. This model is strongly coupled, highly nonlinear, and characterized by physical mechanisms that span a wide range of interacting time scales. Solutions of this system can include very fast component time scales to slowly varying dynamical time scales that are long relative to the normal modes of the model equations. Fully implicit time stepping is attractive for simulating this type of wide-ranging physical phenomena. However, it is essential that one has effective preconditioning strategies so that the overall fully implicit methodology is both efficient and scalable. In this paper, we propose and explore the performance of several candidate block preconditioners for this system. One of these preconditioners is based on an operator-split approximation. This method reduces the

Stabilization and scalable block preconditioning for the Navier-Stokes equations

3\times3

MONOLITHIC MULTIGRID METHODS FOR TWO-DIMENSIONAL RESISTIVE MAGNETOHYDRODYNAMICS ∗

system (momentum, continuity, and magnetics) into two

Approaches for Adjoint-Based A Posteriori Analysis of Stabilized Finite Element Methods

2\times2

A BLOCK PRECONDITIONER FOR AN EXACT PENALTY FORMULATION FOR STATIONARY MHD

operators...

Teko: A block preconditioning capability with concrete example applications in Navier--Stokes and MHD

This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier-Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier-Stokes.

A new class of finite element variational multiscale turbulence models for incompressible 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...

Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation

This study derives a posteriori error estimates for linear functionals of the solution of systems of partial differential equations discretized using stabilized continuous Galerkin methods. We investigate a convection-diffusion equation, the Stokes equations, and incompressible low Reynolds number flow governed by the Navier--Stokes equations. We consider three well-known stabilization methods and show that only one of the three is adjoint consistent and that even this case is contingent upon a proper treatment of the adjoint data. A standard approach for a posteriori error analysis uses the adjoint of the stabilized formulation, which inherits the difficulties induced by the lack of adjoint consistency. We introduce and analyze two alternative approaches. The first is based on the addition of stabilization terms to the adjoint data, while the second is based on a stabilized formulation of the formal adjoint problem. We show that any of the three approaches can be used to derive a fully computable error r...

Towards Extreme-Scale Simulations with Next-Generation Trilinos: A Low Mach Fluid Application Case Study

The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self-adjoint, nonlinear PDEs couples the Navier--Stokes equations for fluids and Maxwells equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time-stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations posed in two-dimensional domains. This formulation has the benefit of implicitly enforcing the divergence-free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techni...