Kolokacioni postupci za rešavanje singularno perturbovanih problema
Collocation methods for solving singular perturbation problems
Author
Radojev, Goran
Mentor
Herceg, DragoslavCommittee members
Zarin, Helena
Herceg, Dragoslav
Teofanov, Ljiljana
Linss, Torsten
Metadata
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U disertaciji su razvijeni kolokacioni postupci sa C1- splajnovima proizvoljnog stepena za rešavanje singularno-perturbovanih problema reakcije-difuzije u jednoj i dve dimenzije. U 1D, pokazano je da kolokacioni postupak sa kvadratnim C1- 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 C1- 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.
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 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.