Weak and Weakened-Weak Formulations

Weak formulations are of fundamental importance for the finite element method (FEM), and naturally, also for meshfree methods. In the FEM, we have basic building blocks of elements, and hence, all the numerical operations including function approximation and integration of the weak forms are all naturally based on the elements. In meshfree methods, however, function approximation and integration are virtually independent, and hence it offers many more innovative ways to use and even establish new weak forms such as the weakened-weak (W2) forms that have unique and important properties, leading to methods that are superior to the standard FEM in many aspects. We first introduce the weak formulation based on the H-space theory and weakened-

weak formulations based on the G-space theory, and discuss some of their important properties. Physical energy principles used for creating weak forms for the FEM and meshfree methods are also outlined in this chapter with emphasis on the novel weakened weak forms. Since both mathematical and physical approaches are used in the literature to establish weak forms, it is sometimes quite confusing to many. This chapter tries to put these two together aiming to show their connections and hence better understand these formulations. Clarity and understanding are often achieved by comparisons. This book gives high preference to the Galerkin weak form for reasons of simplicity,

symmetry, and hence efficiency, which is eventually a crucial factor for any successful numerical method to survive. When we perform all sorts of advances and manipulations, we always try to keep the form of Galerkin, even though our formulation has gone far beyond the standard Galerkin weak form, in terms of implementation, solution=function spaces, and properties. This preference is particularly important for partial differential equations (PDEs) of symmetric operators: we want to preserve the symmetry. We are aware that the contents of this chapter are quite heavy. Readers may skip the

proofs, if their interests are on the applications of these formulation procedures.