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- DIRICHLET PROBLEMS, RAKHIM AITBAYEV
- 2000

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)

- Rakhim Aitbayev
- 2005

Efficient numerical algorithms are developed and analyzed that implementmultilevel preconditioners for the solution of the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner-Fox-Schmit rectangular element and the product two-point Gaussian quadrature. The proposed… (More)

- Rakhim Aitbayev, Bernard Bialecki
- SIAM J. Numerical Analysis
- 2000

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)

- Rakhim Aitbayev
- 2007

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)

- Rakhim Aitbayev, Bernard Bialecki
- SIAM J. Numerical Analysis
- 2003

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)

- R Aitbayev, X C Cai
- 2004

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)

- RAKHIM AITBAYEV
- 2013

We obtain sufficient conditions, easily verifiable, for the existence and uniqueness of piecewise smooth solutions of a linear two-point boundaryvalue problem with general interface conditions. The coefficients of the differential equation may have jump discontinuities at the interface point. As an example, the conditions obtained are applied to a problem… (More)

- Rakhim Aitbayev, Nazgul Yergaliyeva
- Adv. Numerical Analysis
- 2014

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)

- Rakhim Aitbayev
- 2006

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)