PARAMETRIC RAVIART-THOMAS ELEMENTS FOR MIXED METHODS ON DOMAINS WITH CURVED SURFACES

摘要

The finite element approximation on curved boundaries using parametric Raviart-Thomas spaces is studied in the context of the mixed formulation of Poisson's equation as a saddle point system. It is shown that optimal-order convergence is retained on domains with piecewise Ck+2 boundary for the parametric Raviart-Thomas space of degree k >= 0 under the usual regularity assumptions. This extends the analysis in [F. Bertrand, S. Miinzenmaier, and G. Starke, SIAM T. Numer. Anal., 52 (2014), pp. 3165-3180] from the first-order system least squares formulation to mixed approaches of saddle-point type. In addition, a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two-dimensional case. Moreover, the appropriate treatment of inhomogeneous flux boundary conditions is discussed. The results are confirmed by computational results which also demonstrate that optimal-order convergence is not achieved, in general, if standard Raviart-Thomas elements are used instead of the parametric spaces.