Summary: A FAMILY OF RECTANGULAR MIXED
ELEMENTS WITH A CONTINUOUS FLUX
FOR SECOND ORDER ELLIPTIC PROBLEMS
TODD ARBOGAST AND MARY F. WHEELER
SIAM J. NUMER. ANAL. c 2005 Society for Industrial and Applied Mathematics
Vol. 42, No. 5, pp. 19141931
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 -projection operator that is bounded in the Sobolev space H1,
preserves a projection of the divergence, and approximates optimally. Moreover, the corresponding
RaviartThomas flux preserving -projection operator is an L2-projection when restricted to this
family of spaces.
Key words. mixed finite element method, continuous flux, elliptic equation, error estimates
AMS subject classifications. 65N15, 65N30, 35J20
1. Introduction. Mixed finite element methods have been used effectively to
solve many problems, including second order elliptic problems [10, 15, 31, 34]. Both