A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations

We develop the a posteriori error estimation of interior penalty discontinuous Galerkin discretizations for H(curl)-elliptic problems that arise in eddy current models. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The proposed a posteriori error estimator is validated by numerical experiments, illustrating its reliability and efficiency for a range of test problems.

Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular we are able to treat "patchy'" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs. Article published and (c) American Physical Society 2007

Instabilities in threshold-diffusion equations with delay

The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Waves, bumps, and patterns in neural field theories

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.

A bistable reaction-diffusion system in a stretching flow

We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODEs) for the dynamics of the governing advection-reaction-diffusion partial differential equations (PDE), for pulse-like and for plateau-like solutions, based on a non-perturbative approach. This reduction allows us to study the dynamics in two cases: first, close to a saddle-node bifurcation at which a pair of nontrivial steady states are born as the dimensionless reaction rate (Damkoehler number) is increased, and, second, for large Damkoehler number, far away from the bifurcation. The main aim is to investigate the initial-value problem and to determine when an initial condition subject to chaotic stirring will decay to zero and when it will give rise to a nonzero final state. Comparisons with full PDE simulations show that the reduced pulse model accurately predicts the threshold amplitude for a pulse initial condition to give rise to a nontrivial final steady state, and that the reduced plateau model gives an accurate picture of the dynamics of the system at large Damkoehler number. Published in Physica D (2006)

Two-Level Schwarz Preconditioners for Super Penalty Discontinuous Galerkin Methods

We extend the construction and analysis of the non-overlapping Schwarz preconditioners proposed in Antonietti et al. [Math. Model. Numer. Anal., 41(1):21-54, 2007] and [Math. Model. Numer. Anal., submitted, 2006] to the (non-consistent) super penalty discontinuos Galerkin methods introduced by Babuska et al. [SIAM J. Numer. Anal., 10:863-875, 1973] and by Brezzi et al. [Numer. Methods Partial Differential Equations, 16(4):365-378, 2000]. We show that the resulting preconditioners are scalable, and we provide the convergence estimates. We also present numerical experiments demonstrating the theoretical results.