We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to linear systems w...

An analysis is given of particlelike nonlinear bound states in the Newtonian limit of the coupled Einstein-Dirac system introduced by Finster, Smoller and Yau. A proof is given of existence of these bound states in the almost Newtonianian regime, and it is proved that they may be approximated by the energy minimizing solution of the Newton-Schrödinger system obtained by Lieb.

In our work the phenomena determining the performance of semiconductor devices are examined. We use computer simulations based on semi-classical drift-diffusion model. Our aim is to numerically solve a system of three nonlinear elliptic differential equations. For computations, we use a coupled Newton method, with a modification taking advantage of a simple iteration.

Modeling reactive transport in porous media, using a local chemical equilibrium assumption, leads to a system of advection-diffusion PDE’s coupled with algebraic equations. When solving this coupled system, the algebraic equations have to be solved at each grid point for each chemical species and at each time step. This leads to a coupled non-linear system. In this paper a global solution appro...

Newton’s method for the inverse matrix pth root, A−1/p, has the attraction that it involves only matrix multiplication. We show that if the starting matrix is cI for c ∈ R then the iteration converges quadratically to A−1/p if the eigenvalues of A lie in a wedge-shaped convex set containing the disc { z : |z−cp| < cp }. We derive an optimal choice of c for the case where A has real, positive ei...

Stability analysis algorithms coupled with a robust steady state solver are used to understand the behavior of the 2D model problem of thermal convection in a 8 : 1 differentially heated cavity. The system is discretized using a Galerkin=Least Squares Finite Element formulation, and solved to steady state on parallel computers using a fully coupled Newton method and iterative linear solvers. An...

The present report outlines recent results relative to the development of new linear and nonlinear approaches for solving groundwater flow problems. Efficient and robust simulation of complex processes in environmental applications is crucial for understanding, forecasting and performing opportune and reliable decisions. This research effort focuses in developing efficient solvers for two main ...

We compare the performance of the fully coupled Newton–Raphson method with the sequential iteration approach (SIA) for solving the implicit time stepping equations of reactive transport modeling. We formulate the implicit time stepping equations for a demonstration model that incorporates homogeneous equilibrium reactions, i.e. carbonate hydrolysis, and a heterogeneous equilibrium reaction, i.e...

We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantumclassical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrödinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction decomposition. The transport in the classical ...