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Browsing by Author "Kaya, Adem"

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    Doctoral Thesis
    Numerical methods for nonlocal problems
    (Izmir Institute of Technology, 2018-07) Kaya, Adem; Tanoğlu, Gamze
    In this thesis, numerical methods for nonlocal problems with local boundary conditions from the area of peridynamics are studied. The novel operators that satisfy local boundary conditions were proposed as an alternative to the original nonlocal problems which uses nonlocal boundaries. Peridynamic theory is reformulation of continuum mechanics by integral equations for which it has some advantages over traditional partial differential equations. In peridynamic theory, a point can interact with other points within a certain distance which is called horizon and indicated by the parameter δ. In this thesis, we are particularly interested in role of the parameter δ in numerical methods for the novel problems. More precisely, we aim to show its role in condition number, discretization error and convergence factor of multigrid method.
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    Master Thesis
    Pseudo residual-gree bubble functions for the stabilization of convection-diffusion-reaction prolems
    (Izmir Institute of Technology, 2012) Kaya, Adem; Pashaev, Oktay
    Convection - diffusion - reaction problems may contain thin regions in which the solution varies abruptly. The plain Galerkin method may not work for such problems on reasonable discretizations, producing non-physical oscillations. The Residual - Free Bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless in special limit cases. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. In my thesis we will examine a stabilization technique, based on the RFB method and particularly designed to treat the most interesting case of small diffusion in one and two space dimensions for both steady and unsteady convection - diffusion - reaction problems. We replace the RFB functions by their cheap, but efficient approximations which retain the same qualitative behavior. We compare the method with other stabilized methods.