Parametric Raviart--Thomas Elements for Mixed Methods on Domains with Curved Surfaces

Abstract

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 $C^{k+2}$ boundary for the parametric Raviart--Thomas space of degree $k \geq 0$ under the usual regularity assumptions. This extends the analysis in [F. Bertrand, S. Munzenmaier, and G. Starke, SIAM J. 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--Th...