Continuous mixed finite elements for the second order elliptic equation with a low order term

Abstract

We propose a mixed finite element, where the velocity (in terms of Darcy’s law) is approximated by the continuous P k Lagrange elements and the pressure (the prime variable) is approximated by the discontinuous P k − 1 elements, for solving the second order elliptic equation with a low-order term. We show the quasi-optimality for this mixed finite element method. When a low order term is present, the traditional inf–sup condition is no longer required. But the inclusion condition, that the divergence of the discrete velocity space is a subspace of the discrete pressure space, is required. Thus the Taylor–Hood element and most other continuous-pressure mixed elements do not converge. Numerical tests are provided on the new elements and most other popular mixed elements.