Piece-wise Constant Approximations in the Membrane Problem

We present a technique for the approximation of quadratic variational problems of the first order in spaces of piece-wise constant functions. The method adopts ideas from the theory of $\Gamma$-convergence as a guideline, and it differs from more traditional non-conforming techniques because it is based on the introduction of a suitable sequence of discrete functionals to be minimized with no constraints and without requiring that the spline functions fulfill any patch test condition.