The particular solutions of the Laplace equations and their singularities are fundamental
to numerical partial di erential equations in both algorithms and error analysis. We
first review the explicit solutions of Laplace¡¦s equations on sectors with the Dirichlet
and the Neumann boundary conditions. These harmonic functions clearly expose the
solution¡¦s regularity/singularity at the vertex. So we can analyze the singularity of
the Laplace¡¦s solutions on polygons at di erent domain corners and for various boundary
conditions. By using this knowledge we can designed many new testing models
with di erent kind of singularities, like discontinuous and mild singularities, beside the
popular singularity models, Motz¡¦s and the cracked beam problems,
We use the boundary approximation method, i.e. the collocation Tre tz method
in engineering literatures, to solve the above testing models of Laplace boundary value
problems on polygons. Suppose the uniform particular solutions are chosen in the entire
domain. When there is no singularity on all corners, this method has the exponential
convergence. However, its rate of convergence will deteriorate to polynomial if there
exist some corner singularities. From experimental data, we even have three type of
convergence, i.e. exponential, polynomial or their mixed types. We will study these
convergent behaviors and their causes. Finally, we will uncover the relation between
the order of convergence and the intensity of corner singularities.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0717106-100501 |
| Date | 17 July 2006 |
| Creators | Wang, Jau-Ren |
| Contributors | Leevan Ling, Zi-Cai Li, none, Chien-Sen Huang, Tzon-Tzer Lu |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0717106-100501 |
| Rights | withheld, Copyright information available at source archive |
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