Analysis of Generalised Galerkin methods in the numerical solution of elliptic equations

Abstract The Petrov‐Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H1 Galerkin can be interpreted within the Petrov‐Galerkin projection method framework. The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov‐Galerkin numerical process are proposed.