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].