We shall concentrate more on the following Research programmes in the comming three to four years.

- Adaptive Methods for Partial Integro-differential Equations
- Oldroyd Fluids
- H1- Mixed Finite Element Methods
- ADI, Orthogonal Collocation and Qualocation Methods
- Dynamic Model for Particle Size Distribution in Emulsion Polymerization

A crucial basic step in numerical simulation of partial differential equations is to construct reliable and efficient adaptive algorithms with automatic discretization error control within a given tolerance in a given norm.

These algorithms are based on the following two criteria.

- Sharp a posteriori error estimates
- A priori estimates with minimal regularity requirements on the exact solutions (mainly for the dual problems).

This field is wide open and it is certainly waiting for a break through. We hope to take up this issue in future.

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Some Theoretical and Computational Issues. The motion of a Oldroyd fluid gives rise to a non-linear partial integro-differential equation. This can be thought of an integral perturbation of Navier-Stokes equation. Of course, it is possible to discuss the finite element error analysis for smooth solutions. But smoothness requires some compatibility conditions like the initial function should satisfy the Stokes equations at the initial time and hence, it puts unnatural restrictions on the initial data. Attempt will be made to discuss the error estimates with more realistic initial data. In a way, this will generalize the results of Heywood and Rannacher (SIAM J. Numer. Anal. 1982,1986, 1988, 1990).

Recently, there is a spurt of activities on long time behaviour. So theoretical studies like existence of attractors for the Oldroyd equations will be addressed in future. After applying finite element discretization in spatial direction and finite difference scheme in temporal direction, we shall be interested in the convergence of the discret attractor.

In collaboration with Professors Jinyun Yuan and Pedro D. Damázio from the Federal University of Parana at Curitiba (Brazil), we are presently working on the above problem , (ref. [10]-[11] in the list of Submitted Manuscripts).

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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 introduced. Compared to classical methods, the proposed one needs slightly more regularity on the exact solution, but it yields improved estimates for the velocity field ( heat flux) in *L ^{2}* norm.

The above analysis has been extended to a second order hyperbolic equation (with R. K. Sinha and A. K. Otta), refer 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. An extension to parabolic integro-differential equation has been achieved.

Back to TopADI orthogonal collocation methods are found to be very successful for a class of partial differential equations. We shall make an effort to expand thescope to partial integro-differential equations. It will be more challenging to combine these results with efficient quadratures (cf. publ.[14]) while integrating in temporal direction. Some works in collaboration with Professors Graeme Fairweather and Bernard Bialecki are in progress.

The H1-Galerkin method with quadrature (one varient is called qualocation) is successfully applied to the integral as well as boundary integral equations. Recently, results have been derived for two point boundary value problems and some evolution equations in one space dimension. We shall make an effort to extend the analysis to multidimensional problems.

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Emulsion polymerization is a process of great industrial importance, used for the manufacture of latexes for variety of applications such as latex paint, adhesive, coating, binder in paper and textile products and synthetic rubber etc. The mathematical model, developed by Professor Kannan and his group in chemical engg. department in IIT B gives rise to a coupled system of 24 ODE, an integro-differential equation and one first order hyperbolic integro-differential equation. Presently, we are working on the efficient numerical procedures and in future we plan to concentrate not only on the convergence analysis, but also on theoretical investigations like existential analysis and controllability of such systems.

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