Summary: This paper describes a pilot design and implementation of the generalized finite element method (GFEM), as a direct extension of the standard finite element method (SFEM, or FEM), which makes possible accurate solution of engineering problems in complex domains which may be practically impossible to solve using the FEM. The development of the GFEM is illustrated for the Laplacian in two space dimensions in domains which may include several hundreds of voids, and/or cracks, for which the construction of meshes used by the FEM is practically impossible. The two main capabilities are: (1) It is possible to construct the approximation using meshes which may overlap part, or all, of the domain boundary. (2) The method can be incorporated into the approximation handbook functions, which are known analytically, or are generated numerically, and the method can approximate well the solution of boundary value problems in the neighborhood of corner points, voids, cracks, etc.
The main tool is a special integration algorithm, which we call the fast remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporation of the handbook functions into the GFEM is done by employing the partition of unity method. The presented formulations and implementations can be easily extended to the multi-material medium where the voids are replaced by inclusions of various shapes and sizes, and to the elasticity problem. This work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dimensional and nonlinear cases, which are the cases of main interest for future work.