The generalized finite element method: Numerical treatment of singularities, interfaces, and boundary conditions

This dissertation is devoted to numerical approximation of partial differential equations by Generalized Finite Element Method (GFEM), which is closely related to some other methods, such as the hp cloud method and the extended finite element method. As an extension of the standard Finite Element Method (FEM), the GFEM has more flexibility in dealing with complicated domain geometry, corner singularities, transmission problems and mixed boundary conditions. It differs from the standard FEM in the construction of the finite-dimensional space in which the approximate solution is sought. Instead of using piecewise polynomials on each element of a triangulation of the domain, the global GFEM space is defined by using partition of unity in combination with local approximation spaces defined in each patch of the partition. In this sense, the GFEM is an example of so called meshless methods, since the partition of unity need not be subordinated to a particular triangulation of the domain as the standard FEM does. The GFEM allows one to incorporate a priori knowledge of the local behavior of the solution in the construction of the approximation space, and gives the option of constructing trial spaces of any desired regularity.;For transmission (interface) problems on domains with smooth, curved boundaries, we establish quasi-optimal rate of convergence of the numerical solution to the true solution by using a non-conforming GFEM. To achieve this goal, we construct a sequence of approximation spaces Sn, satisfying the following two conditions: (1) nearly zero boundary and interface matching, (2) approximability. We then seek the numerical solution in this spaces as the Galerkin approximation to the true solution, and show that the approximation error of order O( hmn ), where hn is the typical size of elements in the GFEM space Sn, and m is the degree of polynomials used for the local approximation of the solution. Numerical experiments are presented to demonstrate these theoretical results.;We also study the GFEM approximation for Poisson problem in polygonal domains with corner singularities. It is well-known that the loss of regularity of the exact solution due to domain singularities will deteriorate the convergence rate of the standard FEM on quasi-uniform mesh. To circumvent this difficult, we pose the problem in certain weighted Sobolev spaces, and show that the continuous problem has the expected regularity in these spaces. We then construct GFEM approximation spaces using partition of unity and local approximation spaces. For the former, we use dilation techniques to deal with corner singularities, while we use standard piecewise polynomial spaces for the latter. We then establish quasioptimal rate of convergence of the GFEM approximation to the exact solution both in weighted Sobolev spaces and then in Hilbert spaces in terms of O(dim(Sn) --m/2), where dim(Sn) is the dimension of the GFEM space Sn.