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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