SIAM Journal on Numerical Analysis

In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. Our theory is presented entirely in terms of the generalized condition number $\kappa $ of the ...

In two earlier papers [SIAM J. Numer. Anal., 19 (1982), pp. 924–929; 21 (1984), pp. 255–262], we developed an algebraic convergence theory for a class of multigrid methods applied to positive definite self-adjoint linear operator equations. The purpose of ...

We consider a procedure for solving boundary value problems for elliptic homogeneous equations, known as the fundamental solutions method. We prove its applicability for some second order operators as well as for fourth order ones. The boundary conditions ...

A numerical method is described and analyzed for the solution of the exterior Dirichlet problem for the Helmholtz equation in three dimensions. The problem is first reformulated as a Fredholm integral equation of the second kind, based on an approach of ...

In this paper we present and analyze a finite difference scheme for computing approximate solutions and interface curves for the Cauchy problem for the doubly-degenerate parabolic equation $u_t = (u(1 - u)u_x )_x $. Our scheme requires only that linear, ...

Numerically, moving grids are often found to improve the resolution of approximate solutions to nonlinear hyperbolic conservation laws. In this paper we show how to modify many standard numerical flux functions to incorporate this idea. A generalized ...

This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are local error estimates in the sense of the local error in ...

We consider nonlinear problems of the form $f(x,\lambda ,{\bf \alpha }) = 0$, where $x \in \mathbb{R}$ is a state variable, $\lambda \in \mathbb{R}$ is a bifurcation parameter, $\alpha \in \mathbb{R}^p $ is a vector of auxiliary parameters, and f is a ...

We investigate the role of the secant or quasi-Newton condition in the sparse Broyden or Schubert update method for solving systems of nonlinear equations whose Jacobians are either sparse, or can be approximated acceptably by conveniently sparse ...

A new computational test is presented for convergence of Newton-type methods. We obtain componentwise error bounds. The test does not require the second derivative. Numerical examples are given.

In this paper we present interval arithmetic methods for the solution of systems of nonlinear equations. These methods are based on the algorithm of R. Krawczyk [5] and a modification introduced in [3]. Starting with an interval vector containing a ...

Triangular interpolants have been receiving attention recently as a tool for use in free form surface design in computer aided design environments. Traditionally, rectangular patches have been used in most applications; however there are many surfaces ...