On the Fully Nonlinear Alt–Phillips Equation

Entire Solutions of a Fully Nonlinear Equation

C Penalty Methods for the Fully Nonlinear Monge-ampère Equation

In this paper, we develop and analyze C0 penalty methods for the fully nonlinear Monge-Ampère equation det(D2u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well a...

C0 penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C0 penalty methods for the fully nonlinear Monge-Ampère equation det(D2u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well a...

On Fully Discrete Galerkin Methods of Second–order Temporal Accuracy for the Nonlinear Schrödinger Equation

We approximate the solutions of an initialand boundary-value problem for nonlinear Schrödinger equations (with emphasis on the ‘cubic’ nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit, Crank–Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their soluti...

The Nonlinear Schrödinger Equation on the Interval

Let q(x, t) satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, 0 < x < L, with q 0 (x) = q(x, 0), g 0 (t) = q(0, t), f 0 (t) = q(L, t). Let g 1 (t) and f 1 (t) denote the unknown boundary values q x (0, t) and q x (L, t), respectively. We first show that these unknown functions can be expressed in terms of the given initial and bo...