Michael Pilant

A class of finite element methods based on orthonormal, compactly supported wavelets

This paper develops a class of finite elements for compactly supported, shift-invariant functions that satisfy a dyadic refinement equation. Commonly referred to as wavelets, these basis functions have been shown to be remarkably well-suited for integral operator compression, but somewhat more difficult to employ for the representation of arbitrary boundary conditions in the solution of partial differential equations. The current paper extends recent results for treating periodized partial differential equations on unbounded domains in Rn, and enables the solution of Neumann and Dirichlet variational boundary value problems on a class of bounded domains. Tensor product, wavelet-based finite elements are constructed. The construction of the wavelet-based finite elements is achieved by employing the solution of an algebraic eigenvalue problem derived from the dyadic refinement equation characterizing the wavelet, from normalization conditions arising from moment equations satisfied by the wavelet, and from dyadic refinement relations satisfied by the elemental domain. The resulting finite elements can be viewed as generalizations of the connection coefficients employed in the wavelet expansion of periodic differential operators. While the construction carried out in this paper considers only the orthonormal wavelet system derived by Daubechies, the technique is equally applicable for the generation of tensor product elements derived from Coifman wavelets, or any other orthonormal compactly supported wavelet system with polynomial reproducing properties.

An Inverse problem for a nonlinear parabolic equation

On determine une force de contrainte f(u) dependant uniquement de la temperature u(x,t) dans une equation de reaction-diffusion

Spinodal decomposition in a binary liquid mixture

Light scattering was used to study phase separation near the critical temperature Tc in a critical mixture of 2,6‐lutidine and water. This system has an inverted coexistence curve, so that a quench into the two‐phase region is produced by an upward jump in temperature. The scattered intensity I (q,t) was recorded at various angles and at a number of quench depths ΔTf =Tf−Tc in the range 0.5≲ΔTf≲2.5 mK. Here Tf denotes the final temperature. The initial temperature was also varied, and no initial‐state effects were observed. One set of experiments employed a cell of very short optical path (0.1 mm) to minimize multiple scattering at the sacrifice of quenching speed. In another set, emphasis was placed on achieving a temperature jump in roughly 0.1 sec so that phase separation could be followed in its early stages. For ΔTf≲1 mK, the change in the measured ring diameter (qm−1) with time, is in fair agreement with the nonlinear theory of Langer, Bar‐on, and Miller. However, the intensity of the ring, I (qm,t)...

The recovery of potentials from finite spectral data

The reconstruction of a Sturm–Liouville potential from finite spectral data is considered. A numerical technique based on a shooting method determines a potential with the given spectral data. Convergence of reconstructed potentials is shown and numerical examples are considered.

Determining a coefficient in a first-order hyperbolic equation

This paper deals with an inverse problem for a first-order hyperbolic equation: namely, the determination of the coefficient

Estimating parameters in scientific computation - A survey of experience from oil and groundwater modeling

\lambda ( {a,t,\rho } )

An inverse problem for a nonlinear elliptic differential equation

in

An iteration method for the determination of an unknown boundary condition in a parabolic initial-boundary value problem

\rho _t + \rho _a + \lambda \rho = 0

Identification and Control Problems in Petroleum and Groundwater Modeling

. Conditions on the imposed data that lead to a unique solution are presented and an algorithm for the reconstruction of the coefficient is given.

Undetermined Coefficient Problems for Nonlinear Elliptic and Parabolic Equations

and engineering depends upon the accuracy of numerical models. On what does the accuracy of the models depend? Several things: the correctness of the underlying mathematical formulation, the numerical discretization used, the solution algorithms employed, and the accuracy of values for the physical parameters that appear in the model. It is this last factor that we want to examine in some depth. First, what do we mean by a parameter? In physics, a parameter typically describes a property of a material (such as mass, viscosity, or density) that can vary without changing the underlylng natural laws. For example, consider a group of particles moving about in a closed container. The number of particles, their masses, and their charge are parameters. Other parts of a physical model are not parameters. The fact that the equations of motion are given by a sytem of second-order differential equations, for instance, is not describable by a parameter-it is a fundamental characteristic of the model.