A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method

Authors:
Ohannes Karakashian;Charalambos Makridakis
Affiliations:
-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
1999

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Abstract

The convergence of a class of continuous Galerkin methods for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. These methods allow variable temporal stepsizes as well as changing of the spatial grid from one time level to the next. We show the existence of the resulting approximations and prove optimal order error estimates in $L^\infty (L^2 ) $ and in $L^\infty (H^1 ) .$ These estimates are valid under weak restrictions on the space-time mesh. These restrictions are milder if the elliptic projection is used at every time step instead of the L2 projection. We also give superconvergence results at the temporal nodes tn.