**On Convergence of Finite
Difference Methods for Generalized Solutions of Partial Differential Equations**

A programme on the convergence of finite difference approximations to the
generalized solutions of the partial differential and partial
integro-differential equations using finite difference schemes was carried out
during Nov.'90-May'91 in collaboration with S.K. Chung (Seoul National Univ.,
S. Korea) and R.S. Anderssen (CMA,ANU, Canberra) (ref. [3]-[5] in the list of
Research Reports). To motivate the need for the present study, let us consider
a Poisson equation in a square region
say [$\Omega$] in R2 with homogeneous Dirichlet boundary condition. A standard
five point symmetric finite difference scheme may yield O(h2)-order of
convergence with h as the mesh parameter using classical Taylor's Theorem,
provided the solution (say) u is in [$C^4(\Omega)\cap C^3(\bar {\Omega})$]. For
a square domain, one can not expect this order of smoothness for the exact solution. Moreover, the
discontinuity in nonhomogeneous term with this in L2 will imply that the exact
solution uis in H2 which need not be in C2, and in this case, even Taylor's
expansion may not be possible to apply so as to give any meaningful convergence
analysis. But the computational experiments show O(h2)-order of convergence, of
course in the weaker norm like discrete L2-norm. In 1962, this type of question
was raised by Tikhnov and Samarskii and for one dimensional case, they could
come up with some answer. Subsequently, for Poisson equation Makarov as well as
Lazarov could provide some answer. However, even the elliptic problem in
divergence form with variable coefficients the convergence analysis remained
incomplete and attempt has been made by
us to fill this gap. In order to obtain the correct order of
convergence, the present work finds a way out to establish rate of convergence.
The basic techniques used to achieve this goal are Steklov mollification (it is
an averaging process to give greater smoothness)and Bramble-Hilbert Lemma (this
relies heavily on an average Taylor's expansions that is representing a
function in Sobolev spaces in terms of a polynomial and an integral
term-introduced first by S. L. Sobolev for proving the Sobolev Imbedding
Theorems). For the first time, a discrete projection technique is introduced in
the context of finite difference schemes to obtain convergence analysis.

The above analysis is extended to Sobolev equations [18] and [24] in the list
of publications.

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