Gerard R. Richter

An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures

Abstract Numerical methods which are based on boundary integral formulations require the numerical evaluation of the Greens function associated with the problem. In the case of periodic structures, the Greens function is often an infinite series. This infinite series may converge slowly, making numerical evaluation expensive. Here, we present a practical computer implementation of a technique which dramatically speeds up the convergence of the infinite series Greens function associated with the Helmholtz operator. To show the consequences of using this technique, we include some numerical examples.

An Optimal-Order Error Estimate for the Discontinuous Galerkin Method

In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in R2, the method is shown to converge at a rate 0(hn+l) when applied with nth degree polynomial approxi- mations over a semiuniform triangulation, assuming sufficient regularity in the solution.

A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media

The Discontinuous Galerkin (DG) method provides a powerful tool for approximating hyperbolic problems. Here we derive a new space-time DG method for linear time dependent hyperbolic problems written as a symmetric system (including the wave equation and Maxwell’s equations). The main features of the scheme are that it can handle inhomogeneous media, and can be time-stepped by solving a sequence of small linear systems resulting from applying the method on small collections of space-time elements. We show that the method is stable provided the space-time grid is appropriately constructed (this corresponds to the usual time-step restriction for explicit methods, but applied locally) and give an error analysis of the scheme. We also provide some simple numerical tests of the algorithm applied to the wave equation in two space dimensions (plus time).

Numerical identification of a spatially varying diffusion coefficient

We consider the inverse problem of identifying a spatially varying diffusion coefficient on the basis of an observed solution to the forward problem. Under appropriate conditions, this inverse problem can be solved as a first order hyperbolic problem in the unknown coefficient. We provide a modified upwind difference scheme for this hyperbolic problem and prove that its convergence rate is 0(h) when certain conditions are met.

Explicit Finite Element Methods for Symmetric Hyperbolic Equations

A family of explicit space-time finite element methods for the initial boundary value problem for linear, symmetric hyperbolic systems of equations is described and analyzed. The method generalizes the discontinuous Galerkin method and, as is typical for this method, obtains error estimates of order

An explicit finite element method for the wave equation

O(h^{n+1/2})

The discontinuous Galerkin method with diffusion

for approximations by polynomials of degree

Superconvergence of piecewise polynomial Galerkin approximations, for Fredholm integral equations of the second kind

\le n

Analysis of a continuous finite element method for hyperbolic equations

.

Local error estimates for a finite element method for hyperbolic and convection-diffusion equations

Abstract We present a new space—time finite element method for the wave equation. The method is explicit in the sense that, with an appropriate mesh, the finite element approximation can be developed one element at a time. It is thus potentially useful for problems requiring local mesh refinement. Stability and error estimates are derived. The method is shown to be of almost optimal order accuracy in the derivative, or energy, sense. Computational examples are given.