A sharp error estimate for the numerical solution of a Dirichlet problem for the Poisson equation (Q1208564)

The approximate solution of the Poisson equation with Dirichlet boundary conditions on the unit square using the standard five-point difference scheme on a uniform grid with the mesh space \(h\) is considered. For the class of problems for which the exact solution \(y\) satisfies the condition \(y\in C^{(2)}(\overline{\Omega})\) it is shown that the well- known error bounds for \(y_ h-y\), where \(y_ h\) denotes the discrete solution, are sharp. This is shown by means of well-known facts on grid functions, applying the discrete Green function \(G_ h(\xi,\eta)\) to represent the solution of the discrete problem with zero boundary conditions, thus allowing the representation of the error as a function of the exact solution \(y\), and using abstract moduli of continuity with certain properties.