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 Schrodin\-ger 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 $t^\star$. For the blow-up of the amplitude of the solution we recover numerically the well-known rate $(t^\star - t)^{-1/2}$ for d=3. For d=2 our numerical evidence supports the validity of the $\log \log$ law $[\ln\ln \frac {1}{t^\star -t} /(t^\star-t)]^{1/2}$ for t extremely close to $t^\star.$ The scheme also approximates well the details of the blow-up of the phase of the solution at the origin as $t\to t^\star.$

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