<p>U disertaciji su razvijeni kolokacioni postupci sa C¹- splajnovima&nbsp;proizvoljnog stepena za re&scaron;avanje singularno-perturbovanih problema&nbsp;reakcije-difuzije u jednoj i dve dimenzije. U 1D, pokazano je da kolokacioni&nbsp;postupak sa kvadratnim C¹-&nbsp;splajnom na modifikovanoj &Scaron;i&scaron;kinovoj mreži,&nbsp;konvergira uniformno, sa redom konvergencije skoro dva. Takođe, na gradiranim mrežama, ovaj metod ima red konvergencije dva &ndash; uniformno do na logaritamski faktor. Aposterirona ocena je postignuta za kolokacione postupke sa C¹- splajnovima proizvoljnog stepena na proizvoljnoj mreži. Ova ocena je iskori&scaron;ćena i za kreiranje adaptivnih mreža. Numerički rezultati povtrđuju dobijene ocene. U 2D su razmatrane kolokacije sa bikvadratnim splajnovima. Aposterirona ocena gre&scaron;ke je postignuta. Numerički rezultati potvrđuju dobijene teorijske rezultate.<br />&nbsp;</p> / <p>Collocations with arbitrary order C¹-splines for a singularly perturbed reaction-diffusion problem in one dimension and two dimensions are studied. In 1D, collocation with quadratic C¹-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 C¹-splines on arbitrary meshes. These bounds are used to drive an adaptivemeshmoving algorithm. An adaptive algorithm is devised&nbsp;to resolve the boundary layers. Numerical results are presented. In 2D, collocation with biquadratic C¹-spline is studied. Robust a posteriori error bounds are derived for the collocation method on arbitrary meshes. Numerical experiments completed our theoretical results.</p>

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-02-16T11:16:03Z No. of bitstreams: 2 Dissertação - Mayk Joaquim dos Santos - 2017.pdf: 2151565 bytes, checksum: 0afafa6be7f2f9c3ee2a27ca9bf4bf24 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-02-16T11:16:36Z (GMT) No. of bitstreams: 2 Dissertação - Mayk Joaquim dos Santos - 2017.pdf: 2151565 bytes, checksum: 0afafa6be7f2f9c3ee2a27ca9bf4bf24 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-02-16T11:16:36Z (GMT). No. of bitstreams: 2 Dissertação - Mayk Joaquim dos Santos - 2017.pdf: 2151565 bytes, checksum: 0afafa6be7f2f9c3ee2a27ca9bf4bf24 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-01-16 / In this work we will show that, given a piecewise smooth vector field, we can apply the regularization method and, from it, via blow-up, turn it into a singular perturbation problem. By doing that, we can use the tools from singular perturbation theory to perform a qualitative study of piecewise smooth vector fields. Finally, we will show that, through successive changes of coordinates, a singularity of a discontinuous submanifold of codimension k, where k=1 or k=2, can be transformed into a singularity of codimension 0 in order to study the qualitative behavior in this submanifold, where the Filippov’s convention holds. / Neste trabalho mostraremos que, dado um campo de vetores suaves por partes, podemos aplicar o método de regularização e, a partir deste, via “blow-up”, o transformamos em um problema de perturbação singular. Podemos, dessa forma, fazer uso das ferramentas da teoria de perturbação singular para realizar um estudo qualitativo dos campos de vetores suaves por partes. Por último, mostraremos que através de sucessivas mudanças de coordenadas podemos transformar uma singularidade de uma subvariedade de descontinuidade de codimensão k, onde k=1 ou k=2, em uma uma singularidade de codimensão 0 e estudar o comportamento qualitativo ao longo desta subvariedade, onde é válida a convenção de Filippov.