Η ασυνεχή μέθοδο Galerkin για την μη γραμμική εξίσωση του schrodinger στην κρίσιμη διάσταση
Adaptive discontinuous Galerkin finite elements method for the non linear Schrodinger equation
We consider an initial-value problem for the nonlinear Schrodinger with cubic nonlinear-ity in the critical dimension (d = 2). To approximate smooth solutions of this problem we construct and analyse a numerical method where the spatial discretization is based on discontinuous Galerkin finite elements and the temporal discretization is achieved by the implicit Crack-Nickolson scheme. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to ap¬proximate well singular solutions of the NLS equation up to times close to blow-up. The numerical method presented here aims to approximate both radially and non-radially solutions of the NLS.