Rakhim Aitbayev

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We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H1 and H2(More)
We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H1 and H2(More)
A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three-points is required to guarantee optimal order convergence in Sobolev norms. In this article,(More)
We study the computation of the orthogonal spline collocation solution of a linear Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form Lu = ∑ aij(x)uxixj + ∑ bi(x)uxi + c(x)u. We apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a(More)
We discuss our preliminary experiences with several parallel two level additive Schwarz type domain decomposition methods for the simulation of three dimensional transonic compressible ows The focus is on the implementation of the parallel coarse mesh solver which is used to reduce the computational cost and speed up the convergence of the linear algebraic(More)
A quadrature finite element Galerkin scheme for a Dirichlet boundary value problem for the biharmonic equation is analyzed for a solution existence, uniqueness, and convergence. Conforming finite element space of Bogner-Fox-Schmit rectangles and an integration rule based on the two-point Gaussian quadrature are used to formulate the discrete problem. An(More)
A nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton’s method are studied. The mesh independence principle for the collocation problem is proved and used to(More)