An H1 Mixed Finite Element
Method
In order to compute both displacement and velocity accurately, the classical
mixed methods are used in the literature for the last three to four decades.
But the classical mixed methods requires the LBB consistency condition for
the convergence of the numerical
schemes as well as for the wellposedness of the discrete problem and this in
turn puts severe restrictions on the construction of finite element spaces. In
an attempt to avoid this consistency requirement and to use computationally
more attractive piecewise linear spaces for both velocity and displacement
fields (heat flux and temperature distribution in case of heat equation), an
alternate mixed finite element procedure is intriduced in [29] (list of
publications). Compared to classical methods, the proposed one needs slightly more regularity on the exact
solution, but it yields improved estimates or the velocity field ( heat flux)
in L2-norm. We feel strongly that the proposed scheme will open up a series of
new results in future.
The above analysis has been extended to a second order hyperbolic equation
(with R. K. Sinha and A. K. Otta), cf. [7] under the list of Submitted
Manuscripts. Recently, it has been observed by Pani and Fairweather that while
it is difficult to apply the classical mixed method to an evolution equation
with positive type memory term, but the proposed alternate method works well
(cf. [9]). An extension to parabolic integro-differential equation has been
achieved in [10].