scholarly journals High-frequency behaviour of corner singularities in Helmholtz problems

2018 ◽  
Vol 52 (5) ◽  
pp. 1803-1845 ◽  
Author(s):  
T. Chaumont-Frelet ◽  
S. Nicaise

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory.

Author(s):  
Nicolas Delcey ◽  
Philippe Baucour ◽  
Didier Chamagne ◽  
Geneviève Wimmer ◽  
Odile Bouger ◽  
...  

The pantograph strip interface involves many physical phenomena. Temperature evolution is one of them. This problem includes various thermal flux and sources. More specifically, due to the train motion, a moving zigzag heat source occurs. This paper deals with a thermal 2D Alternating Direction Implicit (ADI) numerical method for temperature estimations in the train pantograph carbon strip, the aims being a better wear problems anticipation and the creation of a preventive maintenance. For that, an electrical model is coupled to the thermal one to take into account all Joule effects. The ADI strategy enables a significant computation time reduction against most classical resolution methods. Besides, the model involves two mathematical processes: the first one is an appropriate variable transform which induces a fixed surface heat production, while the second is based on locally refined meshes. Various numerical tests are presented and discussed in order to show the accuracy of the scheme. From a physical point of view, the results are much interesting. Further investigations, depending on the different parameters, should lead us to predict the strip critical thermal phases.


1995 ◽  
Vol 8 (2) ◽  
pp. 177-188
Author(s):  
W. L. Chan ◽  
S. P. Yung

Sharp error estimates for optimality are established for a class of distributed parameter control problems that include elliptic, parabolic, hyperbolic systems with impulsive control and boundary control. The estimates are obtained by constructing manageable dual problems via the extremum principle.


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