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, optimal order error estimates are proved f...

Two piecewise Hermite bicubic orthogonal spline collocation schemes are considered for the approximate solution of elliptic, self-adjoint, nonhomogeneous Dirichlet boundary value problems on rectangles. In the rst scheme, the nonhomogeneous Dirichlet boundary condition is approximated by means of the piecewise Hermite cubic interpolant, while the piecewise cubic interpolant at the boundary Gaus...

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x , α being any real number. The corresponding quasiorthogonal approximation and Gauss-Radau type interpolation are inves...

The increasing process variation and spatial correlation between process parameters makes capacitance extraction of interconnects an important yet challenging problem in modern VLSI designs. In this presentation, we will first introduce basic capacitance extraction flow follow by advance model considering process variation (E.g., orthogonal polynomial method). After that, approaches that solve ...

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 H2-norm error estimate is obtained for th...

Numerical techniques for discretization of velocity space in continuum kinetic calculations are described. An efficient spectral collocation method is developed for the speed coordinate – the radius in velocity space – employing a novel set of non-classical orthogonal polynomials. For problems in which Fokker-Planck collisions are included, a common situation in plasma physics, a procedure is d...

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: Lλ = d2/dx2 − λ2 with constant λ. With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to ...

A model of a three-zone Rapid Thermal Chemical Vapor Deposition (RTCVD) system is developed to study the e ects of spatial wafer temperature patterns and gas-phase reactant depletion on polysilicon deposition uniformity. A sequence of simulated runs is performed, varying the lamp power pro les so that di erent temperature modes are excited. The dominant spatial wafer thermal modes are extracted...

Efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation (OSC) discretization of a Dirichlet boundary value problem with a non–self-adjoint or an indefinite operator. The OSC solution is sought in the Hermite space of piecewise bicubic polynomials. It is proved that the proposed additive an...

In this paper a constrained nonlinear model predictive control (CNMPC) based on deterministic global optimisation is designed. The approach adopted consists in the transformation of the dynamic optimisation problem into a nonlinear programming (NLP) problem using the method of orthogonal collocation on finite elements. Rigorous convex underestimators of the nonconvex NLP problem are then derive...