Numerical methods for hyperbolic and parabolic integro-differential equations

An analysis by energy methods is given for fully discrete numerical methods for time-dependent partial integro-differential equations. Stability and error estimates are derived in H1 and L2. The methods considered pay attention to the storage needs during time-stepping

A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems

Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the published versio

On the existence of maximum principles in parabolic finite element equations

In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem. We then show that for the lumped mass method the above conditions on the triangulation are essentially sharp. This is in contrast to the elliptic case in which the requirements are weaker. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction or a positive operator

Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations

We consider semidiscrete solutions in quasi-uniform finite element spaces of order O(hr) of the initial boundary value problem with Neumann boundary conditions for a second-order parabolic differential equation with time-independent coefficients in a bounded domain in R^N. We show that the semigroup on L∞, defined by the semidiscrete solution of the homogeneous equation, is bounded and analytic uniformly in h. We also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space-time L∞-norm, modulo a logarithmic factor for r = 2, and we give a corresponding almost best approximation property

On positivity and maximum-norm contractivity in time stepping methods for parabolic equations

In an earlier paper the last two authors studied spatially semidiscretepiecewise linear finite element approximations of the heat equation and showed that,in the case of the standard Galerkin method, the solution operator of the initial-valueproblem is neither positive nor contractive in the maximum-norm for small time, butthat for the lumped mass method these properties hold, if the triangulations are essentiallyof Delaunay type. In this paper we continue the study by considering fullydiscrete analogues obtained by discretization also in time. The above properties thencarry over to the backward Euler time stepping method, but for other methods theresults are more restrictive. We discuss in particular the θ-method and the (0; 2) Pad\ue9approximation in one space dimension