The Ritz method with Lagrange multipliers

We develop a general form of the Ritz method for trial functions that do not satisfy the essential boundary conditions. The idea is to treat the latter as variational constraints and remove them using the Lagrange multipliers. In multidimensional problems in addition to the trial functions boundary weight functions also have to be selected to approximate the boundary conditions. We prove convergence of the method and discuss its limitations and implementation issues. In particular, we discuss the required regularity of the variational functional, the completeness of systems of the trial functions, and conditions for consistency of the equations for the trial solutions. The discussion is accompanied by a detailed examination of examples, both analytic and numerical, to illustrate the method.