8,535 research outputs found
Stable Generalized Finite Element Method (SGFEM)
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 the conditioning of the stiffness
matrix of SGFEM 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.Comment: 51 pages, 4 figure
Elements of generalized ultrametric domain theory
Generalized ultrametric spaces are a common generalization of preorders and ordinary ultra-metric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cau
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