James H. Adler

Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations

In this paper, we propose new adaptive local refinement (ALR) strategies for first-order system least-squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration. The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost. Two efficiency measures are discussed: predicted error reduction and predicted computational cost. These methods are first applied to a two-dimensional (2D) Poisson problem with steep gradients, and the results are compared with the threshold-based methods described in [W. Dorfler, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124]. Next, these methods are applied to a 2D reduced model of the incompressible, resistive magnetohydrodynamic equations. These equations are used to simulate instabilities in a large aspect-ratio tokamak. We show that, by using the new ALR strategies on this system, we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance.

First-Order System Least Squares for Incompressible Resistive Magnetohydrodynamics

Magnetohydrodynamics (MHD) is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles. Hence, the equations that describe the plasma form a nonlinear system that couples Navier-Stokes equations with Maxwells equations. This paper shows that the first-order system least squares (FOSLS) finite element method is a viable discretization for these large MHD systems. To solve this system, a nested-iteration-Newton-FOSLS-AMG approach is taken. Most of the work is done on the coarse grid, including most of the linearizations. We show that at most one Newton step and a few V-cycles are all that are needed on the finest grid. Here, we describe how the FOSLS method can be applied to incompressible resistive MHD and how it can be used to solve these MHD problems efficiently. A 3D steady state and a reduced 2D time-dependent test problem are studied. The latter equations can simulate a “large aspect-ratio” tokamak. The goal is to resolve as much physics from the test problems with the least amount of computational work. We show that this is achieved in a few dozen work units or fine grid residual evaluations.

Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics

This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newtons method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most of the iterations being performed on coarser grids, where minimal work is needed to reduce error to the level of discretization error. The approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there. The approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of Newtons method near a solution. The paper develops an algorithm that attempts to optimize accuracy-per-computational-cost on each grid, so that essentially no unnecessary work is done on any grid. The nested iteration algorithm is then applied to a reduced two-dimensional model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Using this algorithm on the MHD equations in the context of a first-order system least squares finite element discretization and algebraic multigrid to solve the linearized systems, instabilities in a model tokamak fusion reactor are simulated. Numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30-80 fine-grid relaxation sweeps per time step.

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...

An Energy-Minimization Finite-Element Approach for the Frank--Oseen Model of Nematic Liquid Crystals

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.

CONSTRAINED OPTIMIZATION FOR LIQUID CRYSTAL EQUILIBRIA

This paper compares the performance of penalty and Lagrange multiplier approaches for the necessary unit-length constraint in the computation of liquid crystal equilibrium configurations. Building on previous work in [SIAM J. Sci. Comput., 37 (2015), pp. S157--S176; SIAM J. Numer. Anal., 53 (2015), pp. 2226--2254], the penalty method is derived and well-posedness of the linearizations within the nonlinear iteration is discussed. In addition, the paper considers the effects of tailored trust-region methods in the context of finite-element discretizations and nested iteration for both formulations. Such methods are aimed at increasing the efficiency and robustness of each algorithms nonlinear iterations. Three representative elastic equilibrium problems are considered to examine each methods performance. The first two configurations have analytical expressions for their exact solutions and, therefore, convergence to the true solution is considered. The third problem considers complicated boundary conditio...

First-order system least squares and the energetic variational approach for two-phase flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.

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

Island Coalescence Using Parallel First-Order System Least Squares on Incompressible Resistive Magnetohydrodynamics

This paper investigates the performance of a parallel Newton, first-order system least-squares (FOSLS) finite-element method with local adaptive refinement and algebraic multigrid (AMG) applied to incompressible, resistive magnetohydrodynamics. In particular, an island coalescence test problem is studied that models magnetic reconnection using a reduced two-dimensional (2D) model of a tokamak fusion reactor. The results show that, using an appropriate temporal and spatial resolution, these methods are capable of resolving the physical instabilities accurately at small computational cost. The time-dependent, nonlinear system of PDEs is solved using work equivalent to about 50--60 simple relaxation sweeps (Gauss--Seidel iterations) per time step. Experiments show that, unless the time step is sufficiently small, nonphysical numerical instabilities may occur. Further, decreasing the time step size does not proportionally increase the cost of the computation, because AMG convergence is improved. In addition, ...