<p>U disertaciji su razvijeni kolokacioni postupci sa C<sup>1</sup>- splajnovima proizvoljnog stepena za rešavanje singularno-perturbovanih problema reakcije-difuzije u jednoj i dve dimenzije. U 1D, pokazano je da kolokacioni postupak sa kvadratnim C<sup>1</sup>- splajnom na modifikovanoj Šiškinovoj mreži, konvergira uniformno, sa redom konvergencije skoro dva. Takođe, na gradiranim mrežama, ovaj metod ima red konvergencije dva – uniformno do na logaritamski faktor. Aposterirona ocena je postignuta za kolokacione postupke sa C<sup>1</sup>- splajnovima proizvoljnog stepena na proizvoljnoj mreži. Ova ocena je iskorišćena i za kreiranje adaptivnih mreža. Numerički rezultati povtrđuju dobijene ocene. U 2D su razmatrane kolokacije sa bikvadratnim splajnovima. Aposterirona ocena greške je postignuta. Numerički rezultati potvrđuju dobijene teorijske rezultate.<br /> </p> / <p>Collocations with arbitrary order C<sup>1</sup>-splines for a singularly perturbed reaction-diffusion problem in one dimension and two dimensions are studied. In 1D, collocation with quadratic C<sup>1</sup>-splines is shown to be almost second order accurate on modified Shishkin mesh in the maximum norm, uniformly in the perturbation parameter. Also, we establish a second-order maximum norm a priori estimate on recursively graded mesh uniformly up to a logarithmic factor in the singular perturbation parameter. A posteriori error bounds are derived for the collocation method with arbitrary order C<sup>1</sup>-splines on arbitrary meshes. These bounds are used to drive an adaptivemeshmoving algorithm. An adaptive algorithm is devised to resolve the boundary layers. Numerical results are presented. In 2D, collocation with biquadratic C<sup>1</sup>-spline is studied. Robust a posteriori error bounds are derived for the collocation method on arbitrary meshes. Numerical experiments completed our theoretical results.</p>
| Identifer | oai:union.ndltd.org:uns.ac.rs/oai:CRISUNS:(BISIS)96047 |
| Date | 22 December 2015 |
| Creators | Radojev Goran |
| Contributors | Herceg Dragoslav, Zarin Helena, Teofanov Ljiljana, Linss Torsten |
| Publisher | Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, University of Novi Sad, Faculty of Sciences at Novi Sad |
| Source Sets | University of Novi Sad |
| Language | Serbian |
| Detected Language | Unknown |
| Type | PhD thesis |
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