A posteriori finite element error estimation for diffusion problems

Adjerid, Babuska, and Flaherty [Math. Models Methods Appl. Sci., 9 (1999), pp. 261-286] and Yu [Math. Numer. Sinica, 13 (1991), pp. 89-101] and [Math. Numer. Sinica, 13 (1991), pp. 307-314] show that a posteriori estimates of spatial discretization errors of piecewise bi-p polynomial finite element solutions of elliptic and parabolic problems on meshes of square elements may be obtained from jumps in solution gradients at element vertices when p is odd and from local elliptic or parabolic problems when p is even. We show that these simple error estimates are asymptotically correct for other finite element spaces. The key requirement is that the trial space contain all monomial terms of degree p + 1 except for x(1)(p+1) and x(2)(p+1) in a Cartesian (x(1); x(2)) frame. Computational results show that the error estimates are accurate, robust, and efficient for a wide range of problems, including some that are not supported by the present theory. These involve quadrilateral-element meshes, problems with singularities, and nonlinear problems.