Error analysis of a continuous-discontinuous galerkin finite element method for generalized 2D vorticity dynamics

摘要

A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method for the generalized two-dimensional vorticity dynamics equations. These equations describe several types of geophysical flows, including the Euler equations. The algorithm consists of a continuous Galerkin finite element method for the stream function and a discontinuous Galerkin finite element method for the ( potential) vorticity. Since this algorithm satisfies a number of invariants, such as energy and enstrophy conservation, it is possible to provide detailed error estimates for this nonlinear problem. The main result of the analysis is a reduction in the smoothness requirements on the vorticity field from H-2(ohm), obtained in a previous analysis, to W-p(r) (ohm) with r > 1/p and p > 2. In addition, sharper estimates for the dependence of the error on time and numerical examples on a model problem are provided.