Collocation methods for solving singular perturbation problems

Abstract

<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>