# Jeff Ovall

## Function, Gradient and Hessian Recovery Using Quadratic Bump Functions

An approximate error function for the discretization error on a given
mesh is obtained by projecting (via the energy inner product) the
functional residual onto the space of continuous, piecewise quadratic
functions which vanish on the vertices of the mesh. Conditions are
given under which one can expect this hierarchical basis error
estimator to give efficient and reliable function recovery,
asymptotically exact gradient recovery and convergent Hessian recovery
in the square norms. One does not find similar function recovery
results in the literature. The analysis given here is based on a
certain superconvergence result which has been used elsewhere in the
analysis of gradient recovery methods. Numerical experiments are
provided which demonstrate the effectivity of the approximate error
function in practice for both the square norms and the absolute value
norms.