A General Purpose Variational Formulation for Boundary Value Problems of Orders Greater than Two

Abstract

We develop a new general purpose variational formulation, particularly suitable for solving boundary value problems of orders greater than two. The functional related to this variational formulation requires only Η1 regularity in order to be well-defined. Using the finite element method based on this new formulation thus becomes simple even for domains in dimensions greater than one.  We prove that a saddle-point solution to the new variational formulation is a weak solution to the associated boundary value problem. We also prove the convergence of the numerical methods used to find approximate solutions to the new formulation. We provide numerical tests to demonstrate the efficacy of this new paradigm.