Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation

We consider the initial-value problem for the radially symmetric nonlinear Schrödinger equation with cubic nonlinearity (NLS) in d = 2 and 3 space dimensions. To approximate smooth solutions of this problem, we construct and analyze a numerical method based on a standard Galerkin finite element spatial discretization with piecewise linear, continuous functions and on an implicit Crank-Nicolson type time-stepping procedure. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the NLS equation that blow up at the origin as the temporal variable t tends from below to a finite value t*. For the blow-up of the amplitude of the solution we recover numerically the well-known rate (t* - t)-1/2 for d = 3. For d = 2 our numerical evidence supports the validity of the log log law [ln ln 1/t*-t/(t* - t)]1/2 for t extremely close to t*. The scheme also approximates well the details of the blow-up of the phase of the solution at the origin as t → t*

Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations

In this paper we present in a unified setting the continuous and discontinuous Galerkin methods for the numerical approximation of the scalar hyperbolic equation. Both methods are stabilized by the interior penalty method, more precisely by the jump of the gradient across element faces in the continuous case whereas in the discontinuous case the stabilization of the jump of the solution and optionally of its gradient is required to achieve optimal convergence. We prove that the solution in the case of the continuous Galerkin approach can be considered as a limit of the discontinuous one when the stabilization parameter associated with the penalization of the solution jump tends to infinity. As a consequence, the limit of the numerical flux of the discontinuous method yields a numerical flux for the continuous method as well. Numerical results will highlight the theoretical results that are proven in this paper

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

International audienceWe derive in this paper a posteriori error estimates for discretizations of convection-diffusion-reaction equations in two or three space dimensions. Our estimates are valid for any cell-centered finite volume scheme, and, in a larger sense, for any locally conservative method such as the mimetic finite difference, covolume, and other. We consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces. We allow for the cases of inhomogeneous and anisotropic diffusion-dispersion tensors and of convection dominance. The estimates are established in the energy (semi-)norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type. They are fully computable (all occurring constants are evaluated explicitly) and locally efficient (give a local lower bound on the error times an efficiency constant), so that they can serve both as indicators for adaptive refinement and for the actual control of the error. They are semi-robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. Numerical experiments confirm their accuracy