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.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be 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 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.

An Indirect Genetic Algorithm for a Nurse Scheduling Problem

This paper describes a Genetic Algorithms approach to a manpower-scheduling problem arising at a major UK hospital. Although Genetic Algorithms have been successfully used for similar problems in the past, they always had to overcome the limitations of the classical Genetic Algorithms paradigm in handling the conflict between objectives and constraints. The approach taken here is to use an indirect coding based on permutations of the nurses, and a heuristic decoder that builds schedules from these permutations. Computational experiments based on 52 weeks of live data are used to evaluate three different decoders with varying levels of intelligence, and four well-known crossover operators. Results are further enhanced by introducing a hybrid crossover operator and by making use of simple bounds to reduce the size of the solution space. The results reveal that the proposed algorithm is able to find high quality solutions and is both faster and more flexible than a recently published Tabu Search approach.

Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.