  Uniform preconditioners for operators of negative order discretized by
(dis)continuous piecewise polynomials of any order are constructed from a
boundedly invertible operator of opposite order discretized by continuous
piecewise linears. Besides the cost of the application of the latter
discretized operator, the other cost of the preconditioner scales linearly with
the number of mesh cells. Compared to earlier proposals, the preconditioner has
the following advantages: It does not require the inverse of a non-diagonal
matrix; it applies without any mildly grading assumption on the mesh; and it
does not require a barycentric refinement of the mesh underlying the trial
space.
