A Family of Rectangular Mixed Elements with a Continuous Flux for Second Order Elliptic Problems

Abstract

We present a family of mixed finite element spaces for second order elliptic equations in two and three space dimensions. Our spaces approximate the vector flux by a continuous function. Our spaces generalize certain spaces used for approximation of Stokes problems. The finite element method incorporates projections of the Dirichlet data and certain low order terms. The method is locally conservative on the average. Suboptimal convergence is proven and demonstrated numerically. The key result is to construct a flux $\pi$-projection operator that is bounded in the Sobolev space H1, preserves a projection of the divergence, and approximates optimally. Moreover, the corresponding Raviart--Thomas flux preserving $\pi$-projection operator is an L2-projection when restricted to this family of spaces.