Donald A. French

Approximation of an elliptic control problem by the finite element method

Finite element approximations of an elliptic control problem are analyzed. Optimal order estimates are obtained in terms of the mesh size h and the target ψ A computational scheme is derived which allows for the application of the conjugate gradient method. The scheme is implemented and tested on some model problems.

A continuous space-time finite element method for the wave equation

We consider a finite element method for the nonhomogeneous second-order wave equation, which is formulated in terms of continuous approximation functions in both space and time, thereby giving a unified treatment of the spatial and temporal discretizations. Our analysis uses primarily energy arguments, which are quite common for spatial discretizations but not for time. We present a priori nodal (in time) superconvergence error estimates without any special time step restrictions. Our method is based on tensor-product spaces for the full discretization.

A space-time finite element method for the wave equation*

A numerical method is introduced which uses finite elements in time and space simultaneously to solve the wave equation. An error analysis for this scheme is presented and the results of several computations. This scheme is accurate for time slabs of arbitrary thickness without introducing extra least squares or stabilization terms.

On the convergence of finite-element approximations of a relaxed variational problem

The accuracy of finite-element approximations to a convex, but not strictly convex, variational problem is considered. Convergence is proved for a finite-element approximation of a particular vector field related to the solution. In a special one-dimensional case,

A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations: part I: The steady state case

0(h)

An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons

convergence is shown for a piecewise linear approximation of the derivative. h denotes the size of each element domain. Numerical results are also presented for this one-dimensional case.

Analysis of a robust finite element approximation for a parabolic equation with rough boundary data

A new upper bound is provided for the L∞-norm of the difference between the viscosity solution of a model steady state Hamilton-Jacobi equation, u, and any given approximation, v. This upper bound is independent of the method used to compute the approximation v; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of v. Numerical experiments investigating the sharpness of the a posteriori error estimate are given.

Numerical Solution of a Thermoviscoelastic Contact Problem by a Penalty Method

It has been suggested that spontaneous synchronous neuronal activity is an essential step in the formation of functional networks in the central nervous system. The key features of this type of activity consist of bursts of action potentials with associated spikes of elevated cytoplasmic calcium. These features are also observed in networks of rat cortical neurons that have been formed in culture. Experimental studies of these cultured networks have led to several hypotheses for the mechanisms underlying the observed synchronized oscillations. In this paper, bursting integrate-and-fire type mathematical models for regular spiking (RS) and intrinsic bursting (IB) neurons are introduced and incorporated through a small-world connection scheme into a two-dimensional excitatory network similar to those in the cultured network.This computer model exhibits spontaneous synchronous activity through mechanisms similar to those hypothesized for the cultured experimental networks. Traces of the membrane potential and cytoplasmic calcium from the model closely match those obtained from experiments. We also consider the impact on network behavior of the IB neurons, the geometry and the small world connection scheme.

Continuous Galerkin finite element methods for a forward-backward heat equation

The approximation of parabolic equations with nonhomogeneous Dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward Euler time discretization is studied. The boundary values are assumed in a least squares sense. It is shown that this method achieves an optimal rate of convergence for rough (only L2) boundary data and for smooth data as well. The results of numerical computations which confirm the robust theoretical error estimates are also presented.

Spatial Distribution of Calcium-Gated Chloride Channels in Olfactory Cilia

We consider the numerical approximation of a one-dimensional quasi-static contact problem in linear thermoviscoelasticity. A finite element approximation based on a penalized problem is proposed and analyzed. We furnish an a priori estimate of the difference between the true and numerical solutions. The results of some computations are also presented.