**Partial Integro-Differential
Equations
**

A programme on numerical methods for both
parabolic and hyperbolic integro-differential equations was carried out in CMA
during Sept'90-Feb'91 in collaboration with Lars B. Wahlbin (Cornell Univ.,
USA) and Vidar Thomée (Chalmers Univ., Sweden). The procedure involved is to
reduce the storage requirements for the fully discrete schemes without loosing accuracy.

Although such a programme was initiated by Thomée and Sloan [SIAM J. Numer.
Anal. 1986], Thomée and Zhang [Math. Comp. 1989] using the spectral arguments
in a crucial way , but the present study would rely on the energy arguments
widening the scope to more general problems (14] in the list of publications).

The effect of numerical quadratures on finite element methods for parabolic
integro-differential equations with nonsmooth initial data was carried out with
Todd Peterson from George Mason University, USA.

The major objective is to see to what extent the known results on parabolic
equations can be carried over to integro-differential equations (ref. [21] in
the list of publications 1 and ref. [8] in the Research Reports).

Subsequently, some improved results were obtained for the nonsmooth case
requiring less regularity on the exact solution [39] in the list of
publications. For parabolic problems, ref. [34]. (with my Ph.D. student Dr.
Rajen K. Sinha). The effect of spatial quadrature on the finite element
approximations to hyperbolic integro-differential equation with minimal
regularity assumptions was carried out with Dr. Sinha (ref. [37] in the list of
publications).

Even for wave equation, the present analysis improves upon the earlier results of Baker et al. (SIAM J. Numer. Anal. 13 (1976)).

It is difficulty to extend the semigroup theoretic arguments of Thomée and
Zhang to time dependent parabolic integro-differential equations with nonsmooth
initial data. Therefore, in [38] and [35] ( list of publications) and Thesis of
R. K. Sinha [3], Pani and Sinha have discussed the finite element error
analysis for the time dependent problem with nonsmooth data using energy
arguments. One of the crucial step in this analysis is to prove he limited
smoothing property and then to use dual problem so as to obtain the desired
results. We note that this settles the open problem cited in the book ‘Finite
Element Methods for Integro-differential Equations', (page 106) written by C.
Chen and T. Shih.

Further, a related question on the negative norm estimates and superconvergence
results is examined in [20] in the list of publications and [10] under Research
Reports.

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