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Energy-Corrected Finite Element Methods for Scalar Elliptic Problems [chapter]

Thomas Horger, Markus Huber, Ulrich Rüde, Christia Waluga, Barbara Wohlmuth
2014 Lecture Notes in Computational Science and Engineering  
In this work, we consider the finite element solution of several scalar elliptic problems with singularities in two dimensions.  ...  We outline recent theoretical developments in energy corrected approaches and demonstrate numerically that by local and easy to implement modifications of the discrete operators, optimal convergence orders  ...  Errors 10 × u − u h (γ) 0,α≈0.96475; Convergence analysis of the energy correction method and standard finite element method for the example with 9 singularities.  ... 
doi:10.1007/978-3-319-10705-9_2 fatcat:hnmp5q5szjcjxmdvlsrmcvzriy

Energy-corrected FEM and explicit time-stepping for parabolic problems [article]

Piotr Swierczynski, Barbara Wohlmuth
2019 arXiv   pre-print
In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation  ...  singular corners and a three-dimensional setting.  ...  Energy-corrected finite element for elliptic equations.  ... 
arXiv:1903.06809v1 fatcat:5c5s3mw435gkjphwpzk2wdggpu

Nested Newton Strategies for Energy-Corrected Finite Element Methods

U. Rüde, C. Waluga, B. Wohlmuth
2014 SIAM Journal on Scientific Computing  
Energy-corrected finite element methods provide an attractive technique for dealing with elliptic problems in domains with re-entrant corners.  ...  In the presence of singular solution components, standard finite element methods on uniform meshes show poor results compared to the best approximation in the L 2 -norm.  ...  In this section, we sketch the idea of energy-corrected finite element methods.  ... 
doi:10.1137/130935392 fatcat:f6nsxrbrnbejzdf3lrbswe7njm

Page 3345 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
Reentrant corners lead to less accurate finite element solutions. The paper mentions a possibility of overcoming this pollution effect.  ...  corners.  ... 

Modeling of field singularities at dielectric edges using grid based methods

C. Classen, E. Gjonaj, U. Römer, R. Schuhmann, T. Weiland
2011 Advances in Radio Science  
This contribution focuses on a modified Finite Integration Technique and on a Discontinuous Galerkin Method with singular approximation functions.  ...  Both methods are able to handle field singularities at perfectly electric conducting as well as at dielectric edges.  ...  Recent developments include the Extended Finite Element Method (XFEM) (Moës et al., 1999) and the Partition of Unity Finite Element Method (PUFEM) (Melenk and Babuška, 1996) .  ... 
doi:10.5194/ars-9-39-2011 fatcat:i6n3ar3q65dfpldcoqy5rymhua

Finite element analyses of three-dimensional crack problems in piezoelectric structures

Fulin Shang, Meinhard Kuna, Martin Abendroth
2003 Engineering Fracture Mechanics  
In this paper, electromechanical fracture mechanics and finite element techniques for crack analyses are extended to three-dimensional crack configurations.  ...  Some errors in the literature concerning the analytical solution for the elliptical crack are corrected.  ...  Rabold for their valuable discussion.  ... 
doi:10.1016/s0013-7944(02)00039-5 fatcat:rdb2volk7vaudpfcctjwvfoynu

Solution Techniques for the Stokes System: A priori and a posteriori modifications, resilient algorithms [article]

Markus Huber, Lorenz John, Petra Pustejovska, Ulrich Rüde, Christian Waluga, Barbara Wohlmuth
2015 arXiv   pre-print
Special local a~priori correction techniques at re-entrant corners lead to an improved representation of the energy in the discrete system and can suppress the global pollution effect.  ...  Local mass conservation can be achieved by an a~posteriori correction to the finite element flux. This avoids artifacts in coupled multi-physics transport problems.  ...  Acknowledgements The authors gratefully acknowledge the support by the Gauss Center for Supercomputing (GCS), the Jülich Supercomputing Centre (JSC), and the German Research Foundation (DFG) through the  ... 
arXiv:1511.05759v1 fatcat:bgkyq3jrhfft5cmyodcma6o24a

Three-dimensional effects on cracked components under anti-plane loading

F. Berto, A. Campagnolo
2015 Frattura ed Integrità Strutturale  
Understanding improved when the existence of corner point singularities and their implications became known.  ...  The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading.  ...  to infinity as a corner point is correct.  ... 
doi:10.3221/igf-esis.33.03 fatcat:3lqayz7tvna5zgwarhhleoyguy

Cracked components under anti-plane loading: recent outcomes and future developments

F. Berto
2017 Frattura ed Integrità Strutturale  
Understanding improved when the existence of corner point singularities and their implications became known.  ...  The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked components under anti-plane loading.  ...  to infinity as a corner point is correct.  ... 
doi:10.3221/igf-esis.41.59 fatcat:xxprs5nrfbbd7jibwpxgqm5dfq

Three-dimensional effects on cracked discs and plates under nominal Mode III loading

A. Campagnolo, F. Berto, L. P. Pook
2015 Frattura ed Integrità Strutturale  
Understanding improved when the existence of corner point singularities and their implications became known.  ...  element (FE) models.  ...  to infinity as a corner point is correct.  ... 
doi:10.3221/igf-esis.34.20 fatcat:jxowcpnjsvdf3ap5qunwftgqz4

Page 819 of Journal of Aircraft Vol. 22, Issue 9 [page]

1985 Journal of Aircraft  
'> Finite element methods in conjunction with special singular crack-tip elements'*!® have also been used to calculate stress intensity factors.  ...  crack lengths c, using the crack-tip finite element method discussed in Ref. 16.  ... 

Eddy Currents and Corner Singularities

François Buret, Monique Dauge, Patrick Dular, Laurent Krahenbuhl, Victor Peron, Ronan Perrussel, Clair Poignard, Damien Voyer
2012 IEEE transactions on magnetics  
Eddy current problems are addressed in a bidimensional setting where the conducting medium is non-magnetic and has a corner singularity.  ...  For any fixed parameter δ linked to the skin depth for a plane interface, we show that the flux density |∇A δ | is bounded near the corner unlike the perfect conducting case.  ...  Fig. 3 . 3 Numerical comparison of the near corner real parts of A δ computed by finite element method and of its approximations given by (8).  ... 
doi:10.1109/tmag.2011.2175378 fatcat:vhkakxvywrhefmiczqykl3weli

The convergence of multilevel methods for solving finite-element equations in the presence of singularities

Harry Yserentant
1986 Mathematics of Computation  
Such triangulations are not suitable for problems with singularities caused by re-entrant corners and abrupt changes in the boundary conditions.  ...  In this paper it is shown that families of properly refined grids yield the same convergence behavior of multi-level methods for such singular problems as quasi-uniform subdivisions do for r72-regular  ...  We want to solve it approximately, using the finite-element method.  ... 
doi:10.1090/s0025-5718-1986-0856693-9 fatcat:2e5bnmfsabdjpjnwy525motnb4

The Convergence of Multi-Level Methods for Solving Finite-Element Equations in the Presence of Singularities

Harry Yserentant
1986 Mathematics of Computation  
Such triangulations are not suitable for problems with singularities caused by re-entrant corners and abrupt changes in the boundary conditions.  ...  In this paper it is shown that families of properly refined grids yield the same convergence behavior of multi-level methods for such singular problems as quasi-uniform subdivisions do for r72-regular  ...  We want to solve it approximately, using the finite-element method.  ... 
doi:10.2307/2008163 fatcat:k5n4zmp6jnh7bhmpwtahmmka4q

Boundary element method based computation of stress intensity factor by modified crack closure integral

S. K. Maiti, N. K. Mukhopadhyay, A. Kakodkar
1997 Computational Mechanics  
Simple relations have been derived for the case of linear, quadratic and quarter point elements around the crack tip. Case studies are presented to demonstrate improvement in the accuracy.  ...  The local smoothing scheme in conjunction with the modified crack closure integral technique has been adopted in the boundary element method to improve the accuracy of computed stress intensity factors  ...  Comparison for SIF correction factor Y for edge crack for quadratic element a/w SIF Correction Factor Y Reference Solution Computed by Displacement Method CCI Method 1st Corner Node 2nd Corner  ... 
doi:10.1007/s004660050169 fatcat:ktfsqi67uve3flrj55flsoc7ma
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