Finite element methods

Tuesday, May 12, 2015 - 3:10pm - 3:35pm
Marcus Sarkis (Worcester Polytechnic Institute)
We present higher-order piecewise continuous finite element methods for solving a class of interface problems where the finite element mesh does not fit the interface. The method is based on correction terms added only to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. We apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.
Tuesday, May 12, 2015 - 2:25pm - 2:50pm
Elizaveta Gordeliy (Schlumberger-Doll Research)
The extended finite element method (XFEM) models crack propagation in a finite-element mesh without domain re-meshing. The discontinuous and singular elastic fields, associated with cracks, are represented by using an enriched shape function space that includes discontinuous and singular functions. This talk presents a fully coupled 2D XFEM model of hydraulic fracture propagation capable of simulating propagation in various regimes including toughness- and viscosity- dominated propagation.
Monday, May 11, 2015 - 3:10pm - 3:35pm
Thomas-Peter Fries (Technische Universität Graz)
The extended finite element method (XFEM) is the method of choice for the simulation of crack propagation. The XFEM enriches the approximation space such that discontinuities and singularities within elements, such as those introduced by cracks, may be captured with optimal accuracy. There is no need to adapt the mesh during the simulation. Herein, the XFEM is used for the simulation of hydraulic fracturing in three spatial dimensions. The crack surface may evolve arbitrarily.
Monday, November 1, 2010 - 10:00am - 10:45am
Douglas Arnold (University of Minnesota, Twin Cities)
The finite element exterior calculus, FEEC, has provided a
viewpoint from which to understand and develop stable finite
element methods for a variety of problems. It has enabled us to
unify, clarify, and refine many of the classical mixed finite
element methods, and has enabled the development of previously
elusive stable mixed finite elements for elasticity. Just as
an abstract Hilbert space framework helps clarify the theory of
finite elements for model elliptic problems, abstract Hilbert
Thursday, October 23, 2014 - 3:40pm - 4:25pm
Daniele Boffi (Università di Pavia)
The analysis of adaptive finite element methods for the approximation of partial differential equations is well established and has been successfully applied to a variety of problems (ranging from source problems to eigenvalue problems).
Thursday, October 23, 2014 - 2:35pm - 3:05pm
Johnny Guzman (Brown University)
The existence of uniformly bounded discrete extension oper
ators is established for conforming Raviart-Thomas and Nedelec elements on locally refined partitions of a polyhedral domain into tetrahedra.
Wednesday, October 22, 2014 - 4:30pm - 5:00pm
Barbara Wohlmuth (Technical University of Munich )
Energy-corrected finite element methods provide an attractive technique to deal with elliptic problems in domains with re-entrant corners. Optimal convergence rates in weighted L2-norms can be fully recovered by a local modification of the stiffness matrix at the re-entrant corner, and no pollution effect occurs. Although the existence of optimal correction factors is established, it is non trivial to determine these factors in practice.
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