George J. Fix

Variational methods for underwater acoustic problems

Abstract Duct acoustic problems differ sharply from the pure exterior problem in that the classical radiation conditions do not prevent wall reflections in the former as they do in the latter. In this paper we derive alternate boundary conditions which do prevent wall reflections, and at the same time can be embedded in a natural way in a Galerkin variational formulation of the duct problem.

On finite element methods of the least squares type

Abstract A theoretical framework for the least squares solution of first order elliptic systems is proposed, and optimal order error estimates for piecewise polynomial approximation spaces are derived. Numerical examples of the least squares method are also provided.

On mixed finite element methods for first order elliptic systems

SummaryA physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the Galerkin finite element solution of these systems. Using this theory in conjunction with a certain hypothesis concerning approximation spaces, optimal error estimates for Galerkin type approximations are demonstrated. An example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided.

A COMPARATIVE STUDY OF FINITE ELEMENT AND FINITE DIFFERENCE METHODS FOR CAUCHY-RIEMANN TYPE EQUATIONS*

A least squares formulation of the system

On the accuracy of least squares methods in the presence of corner singularities

{\operatorname{div}}{\bf u} = \rho

Numerical optimization studies of axisymmetric unsteady sprays

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On finite element approximations of problems having inhomogeneous essential boundary conditions

{\operatorname{curl}}{\bf u} = \zeta

On the Convergence of SOR Iterations for Finite Element Approximations to Elliptic Boundary Value Problems

is surveyed from the viewpoint of both finite element and finite difference ...

Downstream boundary conditions for viscous flow problems

Abstract This paper treats elliptic problems with corner singularities. Finite element approximations based on variational principles of the least squares type tend to display poor convergence properties in such contexts. Moreover, mesh refinement or the use of special singular elements do not appreciably improve matters. Here we show that if the least squares formulation is done in appropriately weighted spaces, then optimal convergence results in unweighted spaces like L2.

On the Finite Element—Least Squares Approximation to Higher Order Elliptic Systems

A hybrid numerical technique is developed for the treatment of axisymmetric unsteady spray equations. An Eulerian mesh is employed for the parabolic gas-phase subsystem of equations while a Lagrangian scheme (or method of characteristics) is utilized for the droplet equations. The integration schemes and the scheme for interpolation between the two meshes are demonstrated to be second-order accurate. The approach is shown to be especially useful in situations where a multivaluedness of the droplet properties occurs due to the crossing of particle paths. A set of model equations are studied but the technique is applicable to a more general and more physically correct set of equations. The effects of interesting numerical parameters such as mesh size, number of droplet characteristics, time step, and the injection pulse time are determined via a parameter study. In addition to confirming quadratic convergence, the results indicate slightly more sensitivity to grid spacing than to the number of characteristics.