Daniele Funaro

Polynomial approximation of differential equations

Special Families of Polynomials.- Orthogonality.- Numerical Integration.- Transforms.- Functional Spaces.- Results in Approximation Theory.- Derivative Matrices.- Eigenvalue Analysis.- Ordinary Differential Equations.- Time-Dependent Problems.- Domain-Decomposition Methods.- Examples.- An Example in Two Dimensions.

An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods

A domain decomposition method for second-order elliptic problems is considered. An iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed. At each iteration, a relaxation is accomplished at the subdomain interfaces. In several circumstances, a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found. Moreover, when such a value is not available, an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed.This iterative method is then applied to the spectral collocation approximation of the differential problem. The same kind of convergence results are proven. Many numerical experiments show the effectiveness of the method proposed here.

Approximation of some diffusion evolution equations in unbounded domains by Hermite functions

Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.

A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

The Schwarz algorithm for spectral methods

Recently, the Schwarz alternating method has been successfully coupled to spatial discretizations of spectral type, in order to solve boundary value, problems in complex, geometries with infinite order accuracy. In this paper, a simple version of the method is considered. A proof of its convergence is given in the energy norm, exploiting the properties of discrete-harmonic polynomials and a discrete maximum principle for spectral methods. More general situations can be handled theoretically in one space dimension.

Spectral Elements for Transport-Dominated Equations

1. The Poisson equation in the square.- 2. Steady transport-diffusion equations.- 3. Other kinds of boundary conditions.- 4. The spectral element method.- 5. Time discretization.- 6. Extensions.- References.

A Preconditioning Matrix for the Chebyshev Differencing Operator

An efficient preconditioner for the Chebyshev differencing operator is considered. The corresponding preconditioned eigenvalues are real and positive and lie between 1 and

Some results about the pseudospectral approximation of one-dimensional fourth-order problems

{\pi / 2}

A new scheme for the approximation of advection-diffusion equations by collocation

. An eixpelicit formula for these eigenvalues and the corresponding eigenfunctions is given. The results are generalized to the case of operators related to Chebyshev discretizations of systems of linear differential equations.

Domain Decomposition Methods for Pseudo Spectral Approximations. Part I. Second Order Equations in one Dimension.

SummaryWe analyze the pseudospectral approximation of fourth order problems. We give convergence results in the one dimensional case. Numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem. Eigenvalues and preconditioning are also investigated.