#### SFEM (Strong Formulation Finite Element Method)

F. Tornabene

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SFEM is an acronym for Strong Formulation Finite Element Method. The SFEM is a numerical procedure that decomposes the physical domain or problem in finite elements and uses the strong formulation inside each element mapped on the parent (or computational) element. When in the above procedure the weak formulation is used (instead of the strong form), the WFEM is defined. The latter is well-known in literature as FEM. Generally, practical engineering problems are complex due to geometry, material and load discontinuities. For solving them, it is necessary to divide the whole domain into finite elements of arbitrary shape. The mapping technique is introduced at this level to transform an arbitrary shaped element to a parent element (computational element). The classic Finite Element Method (FEM) uses the above mentioned procedures and the problem at the parent element level is solved using the weak (variational) formulation. In this sense the method in hand can be named Weak Formulation Finite Element Method (WFEM). Analogously, following the previously reported scheme, the same differential problem involving the parent element can be solved using the strong formulation. This new approach can be called Strong Formulation Finite Element Method (SFEM). As a matter of fact, the domain division (finite elements) and the mapping are common to WFEM and SFEM. The most significant difference between these two methodologies lays on the formulation used for solving the parent element. In other words, the letters ‘W’ and ‘S’ are related to the formulation used at the parent element level, whereas ‘FEM’ is associated to the division in finite elements of the physical domain.

The aim of this method is to have a general view on collocation methods, which are based on the strong formulation of a mathematical problem. The general view of the present technique makes the method understandable to the readers, especially the ones who are keen on weak formulation methodologies.The accuracy of the SFEM can be improved in two ways. The classic way is related to the mesh refinement when the same degree of the elements is fixed: this approach is known as h-version of FEM. The second way keeps the mesh size and increases the approximating polynomial functions: this method is called p-version of FEM. The hybrid approach hp-version has also been presented. In short, the SFEM can be seen as an hp-version of FEM when strong formulation is solved, instead of the weak one as in FEM.