In the begining of my talk, I plan to answer the question `Why there is a relook at Mathematics in the last 60 to seventy years ?' Apart from traditional Pure and Applied Mathematics, a new branch of Mathematics called Computational Mathematics (GenerallyScientific Computing) has emerged. I shall focus on Computational PDE, a part of Scientific Computing as it is close to my heart. It is important to know `what are major objectives or emphasis in Computational PDE?' Then I shall take a toy problem like a simple Poission Problem and talk
about `why we resort to numerical approximations ?' after describing various numerical methods, I shall devote my time on finite difference methods (FDM) applied to this toy problem in a square. My focus is to show that how theory of PDE helps to articulate the questions in Computational PDE and then discuss stabilty and convergence analysis. Finally, I shall touch upon some important questions posed by this simple toy problem when finite difference method is applied to it and try to provide some answers.