Shirley B. Pomeranz

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.

An asymptotically exact finite element error estimator based on C1 stress recovery

A posteriori finite element error estimation for planar linear elasticity problems is developed using a recovery operator based on a C1 stress smoothing technique developed by Tessler, Riggs and Macy. The error estimator that is developed is proved to be asymptotically exact under reasonable regularity assumptions on the mesh and the solution. Numerical results for a typical plane stress problem are given.

Computational methods for nonlinear elliptic eigenvalue problems

Abstract In this paper, methods for finding nontrivial solutions of the nonlinear eigenvalue problem − δu = λF ′( u ) are considered. An equivalent variational formulation is used to obtain an iterative procedure for solving the problem with a general (non-convex) function F ( u ). The global convergence of this procedure is established, i.e., convergence from any initial guess. The method is applied to a test problem with F ( u ) = − cos u .

Aitken’s Δ2 method extended

Aitken’s method is used to accelerate convergence of sequences, e.g. sequences obtained from iterative methods. An explicit assumption in deriving Aitken’s method and establishing acceleration (for linearly convergent sequences) is that consecutive error iterates (or their approximations) have the same sign or have an alternating sign pattern. We extend the standard Aitken’s method to the cases in which consecutive pairs of error iterates in the sequence have alternating signs. Under suitable restrictions, acceleration of convergence is proved. Implementation of our extended method is described. Numerical examples demonstrate the process. An example is included relating our results to results obtained from Richard extrapolation.

A Note on Transforming a Plane Strain First-Kind Fredholm Integral Equation into an Equivalent Second-Kind Equation

Methods to convert Fredholm integral equations of the first kind into equivalent Fredholm integral equations of the second kind are used to study issues of existence and uniqueness of solutions. For some examples applied to plane strain problems, see (Constanda, Proc. Amer. Math. Soc. 123:3385–3396, 1995) and (Constanda, Direct and Indirect Boundary Integral Equations. Chapman & Hall/CRC, Boca Raton-London-New York-Washington, DC, 2000, Sec. 2.12). In this paper, another technique to convert the Fredholm integral equation of the first kind that arises in a direct boundary integral formulation for the plane strain Dirichlet problem into an equivalent Fredholm integral equation of the second kind is developed. The technique presented in this paper generalizes work of Y. Yan and I.H. Sloan that was done for the scalar Laplace equation (Yan and Sloan, J. Integral Equations Appl. 1:549–579, 1988) to the plane strain system of displacement equations.

A Contact Problem for a Convection-diffusion Equation

We have proposed an efficient iterative domain decomposition method that solves general convection-diffusion singular perturbation problems. Our specific application involves a piecewise constant diffusion coefficient. We have established sufficient conditions for the convergence of the method, identified suitable values of the relaxation parameter ϑ, and started investigations into the rate of convergence. The method was implemented to O(e2) to solve test problems.

The Nature of Mathematical Modeling. By Neil Gershenfeld

(2000). The Nature of Mathematical Modeling. By Neil Gershenfeld. The American Mathematical Monthly: Vol. 107, No. 8, pp. 763-766.

Construction of some iterative methods for solving boundary element linear systems

Abstract Some general types of iterative methods for solving linear systems of algebraic equations are discussed. The methods are applied to linear systems, as arising from boundary element methods, in which known and unknown components of a vector are treated together. The structure of the vector is to be preserved. Some SOR-type and conjugate gradient-type iterative methods are proposed and compared. The effect of the locations of the known components of the solution vector on the rate of convergence is also investigated. Results demonstrate that the conjugate gradient-type methods can be superior to the SOR-type methods with respect to rate of convergence.

A solitary wave application of an iterative method for nonlinear elliptic eigenvalue problems

Abstract A basic problem is that of solving nonlinear elliptic eigenvalue problems in the plane. Problems of this type arise in many branches of physics and engineering. The particular problem considered here, internal solitary waves in stratified fluids, occurs in fluid dynamics. An iterative technique (based on a transformation of the objective functional in an equivalent variational formulation of the problem) is used to numerically solve for eigenvalues and eigenfunctions associated with the fluid system. The choice of implementation involves discretizing the problem using finite Fourier sine series in conjunction with fast Fourier transforms (FFT). Results from preliminary test problems indicate that this numerical method can be used to verify the range of validity of existing (analytic) asymptotic expansion methods, and that new large amplitude solutions (strongly nonlinear waves) that cannot be obtained from the existing asymptotic expansion methods can be obtained using this numerical method.