Peyman Hessari

A modified symmetric successive overrelaxation method for augmented systems

In this paper, we establish a modified symmetric successive overrelaxation (MSSOR) method, to solve augmented systems of linear equations, which uses two relaxation parameters. This method is an extension of the symmetric SOR (SSOR) iterative method. The convergence of the MSSOR method for augmented systems is studied. Numerical examples show that the new method is an efficient method.

The least-squares pseudo-spectral method for Navier-Stokes equations

A spectral collocation approximation of first-order system least squares for incompressible Stokes equations was analyzed in Kim et al. (2004) [12], and finite element approximations for incompressible Navier-Stokes equations were developed in Bochev et al. (1998,1999) [9,10]. The aim of this paper is to analyze the first-order system least-squares pseudo-spectral method for incompressible Navier-Stokes equations. The paper will be an extension of the result in Kim et al. (2004) [12] to the Navier-Stokes equations. Our least-squares functional is defined by the sum of discrete spectral norms of a first-order system of equations corresponding to the Navier-Stokes equations based on Legendre-Gauss-Lobatto points. We show its ellipticity and continuity over an appropriate product space, and spectral convergences of discretization errors are derived in the H^1-norm and the L^2-norm in each variable. Finally, we present some numerical examples.

First order system least squares method for the interface problem of the Stokes equations

The first order system least squares method for the Stokes equation with discontinuous viscosity and singular force along the interface is proposed and analyzed. First, interface conditions are derived. By introducing a physical meaningful variable such as the velocity gradient, the Stokes equation transformed into a first order system of equations. Then the continuous and discrete norm least squares functionals using Legendre and Chebyshev weights for the first order system are defined. We showed that continuous and discrete homogeneous least squares functionals are equivalent to appropriate product norms. The spectral convergence of the proposed method is given. A numerical example is provided to support the method and its analysis.

PRECONDITIONED SPECTRAL COLLOCATION METHOD ON CURVED ELEMENT DOMAINS USING THE GORDON-HALL TRANSFORMATION

The spectral collocation method for a second order elliptic boundary value problem on a domain with curved boundaries is stud- ied using the Gordon and Hall transformation which enables us to have a transformed elliptic problem and a square domain S = (0,h) × (0,h), h > 0. The preconditioned system of the spectral collocation approx- imation based on Legendre-Gauss-Lobatto points by the matrix based on piecewise bilinear finite element discretizations is shown to have the high order accuracy of convergence and the efficiency of the finite element preconditioner.

Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

Analysis of least squares pseudo-spectral method for the interface problem of the Navier-Stokes equations

The aim of this paper is to propose and analyze the first order system least squares method for the incompressible Navier-Stokes equation with discontinuous viscosity and singular force along the interface as the earlier work of the first author on Stokes interface problem (Hessari, 2014). Interface conditions are derived, and the Navier-Stokes equation transformed into a first order system of equations by introducing velocity gradient as a new variable. The least squares functional is defined based on L 2 norm applied to the first order system. Both discrete and continuous least squares functionals are put into the canonical form and the existence and uniqueness of branch of nonsingular solutions are shown. The spectral convergence of the proposed method is given. Numerical studies of the convergence are also provided.

Pseudo-spectral least squares method for linear elasticity

Abstract The first order system least squares Legendre and Chebyshev spectral method for two dimensional space linear elasticity is investigated. The drilling rotation is defined as a new variable and the linear elasticity equation is supplemented with an auxiliary equation. The weighted L 2 -norm least squares principle is applied to a stress–displacement–rotation. It is shown that the homogeneous least squares functional is equivalent to weighted H 1 -norm like for stress and weighted H 1 -norm for displacement and rotation. This weighted H 1 -norm equivalence is λ -uniform. Spectral convergence for both Legendre and Chebyshev approaches are given along with some numerical experiments. The generalization for three dimensional spaces is also provided.

Convergence analysis of preconditioned AOR iterative method

In this paper, we consider a preconditioned AOR (PAOR) method to solve systems of linear equations. We show the convergence of the PAOR method. We also give comparison results when the coefficient matrix is an L- or H-matrix. Finally, we provide some numerical experiments to show efficiency of PAOR method.