The Maximum Angle Condition for Mixed and Nonconforming Elements: Application to the Stokes Equations

Abstract

For the Lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition. The objective of this paper is to obtain similar results for the Raviart--Thomas interpolation arising in the analysis of mixed methods. We prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and, moreover, that this condition is indeed necessary to have these estimates. Error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the Stokes equations are derived from our results.