In this talk, we combine the FOSLS method with the FOSLL
method to
create a Hybrid method.
The FOSLS approach minimizes the error,
, over a finite element
subspace,
,
in the operator norm,
. The FOSLL
method
looks for an approximation
in the range of
, setting
and choosing
, a
standard finite element space.
FOSLL
minimizes the
norm of the error over
, that is,
.
FOSLS enjoys a locally sharp, globally reliable, and easily computable a posterior error
estimate, while FOSLL
does not.
The hybrid method attempts to retain the best properties of both FOSLS and FOSLL
.
This is accomplished by combining the FOSLS functional, the FOSLL
functional, and an
intermediate term that draws them together. The Hybrid method produces an approximation,
, that is nearly the optimal over
in the graph norm,
.
The FOSLS and intermediate terms in the Hybrid functional provide a very effective a
posteriori error measure.
We show that the hybrid functional is coercive and continuous in the
graph-like
norm with modest constants,
and
; that both
and
converge with rates based on standard interpolation bounds; and that if
has full
regularity,
the
error,
, converges with a full power of the discretization
parameter,
, faster
than the functional norm. Letting
denote the optimum over
in the
graph norm, we also
show that if superposition is used, then
converges two powers of
faster than the functional norm. Numerical tests are provided
to confirm the efficiency of the Hybrid method.