Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods

Magneto-Roton Modes of the Ultra Quantum Crystal: Numerical Study

The Field Induced Spin Density Wave phases observed in quasi-one-dimensional conductors of the Bechgaard salts family under magnetic field exhibit both Spin Density Wave order and a Quantized Hall Effect, which may exhibit sign reversals. The original nature of the condensed phases is evidenced by the collective mode spectrum. Besides the Goldstone modes, a quasi periodic structure of Magneto-Roton modes, predicted to exist for a monotonic sequence of Hall Quantum numbers, is confirmed, and a second mode is shown to exist within the single particle gap. We present numerical estimates of the Magneto-Roton mode energies in a generic case of the monotonic sequence. The mass anisotropy of the collective mode is calculated. We show how differently the MR spectrum evolves with magnetic field at low and high fields. The collective mode spectrum should have specific features, in the sign reversed "Ribault Phase", as compared to modes of the majority sign phases. We investigate numerically the collective mode in the Ribault Phase.Comment: this paper incorporates material contained in a previous cond-mat preprint cond-mat/9709210, but cannot be described as a replaced version, because it contains a significant amount of new material dealing with the instability line and with the topic of Ribault Phases. It contains 13 figures (.ps files

Asymptotically exact a posteriori error estimates for the BDM finite element approximation of mixed Laplace eigenvalue problems

We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the eigenfunction of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the divergence without compromising the approximation properties. Consequently, the new flux can be used to derive optimal and asymptotically exact upper bounds for the eigenfunction, and optimal upper bounds for the corresponding eigenvalue. Numerical examples validate the theory and motivate the use of an adaptive mesh refinement.</p

Multiple sclerosis, the measurement of disability and access to clinical trial data

Background: Inferences about long-term effects of therapies in multiple sclerosis (MS) have been based on surrogate markers studied in short-term trials. Nevertheless, MS trials have been getting steadily shorter despite the lack of a consensus definition for the most important clinical outcome - unremitting progression of disability. Methods: We have examined widely used surrogate markers of disability progression in MS within a unique database of individual patient data from the placebo arms of 31 randomised clinical trials. Findings: Definitions of treatment failure used in secondary progressive MS trials include much change unrelated to the target of unremitting disability. In relapsing-remitting MS, disability progression by treatment failure definitions was no more likely than similarly defined improvement for these disability surrogates. Existing definitions of disease progression in relapsing-remitting trials encompass random variation, measurement error and remitting relapses and appear not to measure unremitting disability. Interpretation: Clinical surrogates of unremitting disability used in relapsing -remitting trials cannot be validated. Trials have been too short and/or degrees of disability change too small to evaluate unremitting disability outcomes. Important implications for trial design and reinterpretation of existing trial results have emerged long after regulatory approval and widespread use of therapies in MS, highlighting the necessity of having primary trial data in the public domain

PARADOXES OF INFORMATION SYSTEMS PLANNING

During two research projects to identify the difficulties associated with information systems planning (ISP), it became apparent that IS managers and users often do not apply a number of commonly accepted guidelines for successful ISP. This paper reports some instances in which the guidelines were not applied. It also explains the neglect of the guidelines and the incentives ISP participants had for not applying them. The findings suggest further research and also have practical implications for IS managers

Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media

We demonstrate the existence of stable three-dimensional spatiotemporal solitons (STSs) in media with a nonlocal cubic nonlinearity. Fundamental (nonspinning) STSs forming one-parameter families are stable if their propagation constant exceeds a certain critical value, that is inversely proportional to the range of nonlocality of nonlinear response. All spinning three-dimensional STSs are found to be unstable.Comment: 14 pages, 6 figures, accepted to PRE, Rapid Communication

A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators

The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a point-wise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. (SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods

A Decision Support System for Student Transfer Advising

Many students start their academic careers at community colleges. After a year or two, they transfer to a university to complete their undergraduate degrees. Students who make poor course selections at community colleges may find that some of their course credits do not transfer to the university programs of their choice. A decision support system was developed to help community college students (1) understand the structure of university undergraduate degree programs, and (2) identify community college courses that meet university requirements. The system is designed to be easy to use and attractive for the typical community college student. It was consiructed by a team of three undergraduate MIS students over the course of a single semester using a Windows-based hypermedia tool