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Abstract: The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that SGFEM retains the excellent convergence properties of GFEM, does not require a ramp-function in the presence of blending elements, and the conditioning of the associated stiffness matrix is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples. [Copyright &y& Elsevier]

In a recent paper, the authors introduced the recovery method (local energy matching principle) for solving large systems of lattice equations. The idea is to construct a partial differential equation along with a finite element discretization such that the arising system of linear equations has equivalent energy as the original system of lattice equations. Since a vast variety of efficient solvers is available for solving large systems of finite element discretizations of elliptic PDEs, these solvers may serve as preconditioners for the system of lattice equations. In this paper, we will focus on both the theoretical and the numerical dependence of the method on various mesh-dependent parameters, which can be easily computed and monitored during the solution process. Systematic parameter tests have been performed which underline (a) the robustness and the efficiency of the recovery method and (b) the reliability of the control parameters, which are computed in a preprocessing step to predict the performance of the preconditioner based on the recovery method. [ABSTRACT FROM AUTHOR]

Interior estimates are proved in the L∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations. [ABSTRACT FROM AUTHOR]

The pollution problem of the p-version of the finite element method is studied. It is shown that the pollution effect can be practically dealt with by a proper mesh design. [ABSTRACT FROM AUTHOR]

In this paper, we present an extension of the Stable Generalized FEM (SGFEM) for 3-D fracture mechanics problems. Numerical experiments show that the use of available singular enrichment bases derived from the 2-D asymptotic crack tip solution leads to a severe loss of accuracy in the SGFEM. An enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the SGFEM. The proposed SGFEM for 3-D fractures delivers the same rate of growth in condition number as the standard FEM for proper choice of singular enrichment functions. The accuracy and conditioning obtained with the SGFEM is compared with the Generalized FEM (GFEM) for different types of singular enrichment bases. A fully 3-D fracture mechanics problem is considered to highlight issues that arise only in 3-D problems. The convergence of Stress Intensity Factors (SIFs) extracted from GFEM and SGFEM solutions and the effect of the size of enrichment sub-domains on the accuracy of extracted SIFs are also studied. It is shown that the accuracy of SIFs may deteriorate with mesh refinement when the topological enrichment strategy is adopted. The proposed SGFEM delivers much better accuracy than the GFEM for both geometrical and topological enrichment strategies. [ABSTRACT FROM AUTHOR]

Highlights: [•] Conditioning of stiffness matrix in the SGFEM grows at the same rate as in the FEM. [•] SGFEM with linear Heaviside enrichments yields optimal convergence rates. [•] Rules for selection of optimal set of enrichment nodes for the SGFEM are presented. [•] The proposed SGFEM for 2-D fractures is significantly more accurate than the GFEM. [•] A simple and yet generic implementation strategy for the SGFEM is described. [Copyright &y& Elsevier]

Abstract: In this paper, we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy. The main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution. This is often imposed by the size of the problem domain, the capabilities for mesh generation, and the employed finite element software and cannot be changed. Hence, there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non-intrusive approaches with respect to the employed software. [Copyright &y& Elsevier]