Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article, we develop the a priori and a posteriori error analysis of hp-version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ ℝd, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp-adaptive refinement algorithm.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

Essential spectrum of systems of singular differential equations

In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

Differential healing response attributed to culprit lesions of patients with acute coronary syndromes and stable coronary artery after implantation of drug-eluting stents: An optical coherence tomography study

BACKGROUND Pathology studies have shown delayed arterial healing in culprit lesions of patients with acute coronary syndrome (ACS) compared with stable coronary artery disease (CAD) after placement of drug-eluting stents (DES). It is unknown whether similar differences exist in-vivo during long-term follow-up. Using optical coherence tomography (OCT), we assessed differences in arterial healing between patients with ACS and stable CAD five years after DES implantation. METHODS AND RESULTS A total of 88 patients comprised of 53 ACS lesions with 7864 struts and 35 stable lesions with 5298 struts were suitable for final OCT analysis five years after DES implantation. The analytical approach was based on a hierarchical Bayesian random-effects model. OCT endpoints were strut coverage, malapposition, protrusion, evaginations and cluster formation. Uncovered (1.7% vs. 0.7%, adjusted p=0.041) or protruding struts (0.50% vs. 0.13%, adjusted p=0.038) were more frequent among ACS compared with stable CAD lesions. A similar trend was observed for malapposed struts (1.33% vs. 0.45%, adj. p=0.072). Clusters of uncovered or malapposed/protruding struts were present in 34.0% of ACS and 14.1% of stable patients (adj. p=0.041). Coronary evaginations were more frequent in patients with ST-elevation myocardial infarction compared with stable CAD patients (0.16 vs. 0.13 per cross section, p=0.027). CONCLUSION Uncovered, malapposed, and protruding stent struts as well as clusters of delayed healing may be more frequent in culprit lesions of ACS compared with stable CAD patients late after DES implantation. Our observational findings suggest a differential healing response attributable to lesion characteristics of patients with ACS compared with stable CAD in-vivo.

Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a stateof- the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

A boundary value problem approach to too-short arc optical track association

Given a short-arc optical observation with estimated angle-rates, the admissible region is a compact region in the range / range-rate space defined such that all likely and relevant orbits are contained within it. An alternative boundary value problem formulation has recently been proposed where range / range hypotheses are generated with two angle measurements from two tracks as input. In this paper, angle-rate information is reintroduced as a means to eliminate hypotheses by bounding their constants of motion before a more computationally costly Lambert solver or differential correction algorithm is run.

Fully Adaptive Newton--Galerkin Methods for Semilinear Elliptic Partial Differential Equations

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

Computational Comparison of Continuous and Discontinuous Galerkin Time-Stepping Methods for Nonlinear Initial Value Problems

This article centers on the computational performance of the continuous and discontinuous Galerkin time stepping schemes for general first-order initial value problems in R n , with continuous nonlinearities. We briefly review a recent existence result for discrete solutions from [6], and provide a numerical comparison of the two time discretization methods.

Cognitive-Behavioural Analysis System of Psychotherapy (CBASP), a drug, or their combination: differential therapeutics for persistent depressive disorder: a study protocol of an individual participant data network meta-analysis.

INTRODUCTION Despite important advances in psychological and pharmacological treatments of persistent depressive disorders in the past decades, their responses remain typically slow and poor, and differential responses among different modalities of treatments or their combinations are not well understood. Cognitive-Behavioural Analysis System of Psychotherapy (CBASP) is the only psychotherapy that has been specifically designed for chronic depression and has been examined in an increasing number of trials against medications, alone or in combination. When several treatment alternatives are available for a certain condition, network meta-analysis (NMA) provides a powerful tool to examine their relative efficacy by combining all direct and indirect comparisons. Individual participant data (IPD) meta-analysis enables exploration of impacts of individual characteristics that lead to a differentiated approach matching treatments to specific subgroups of patients. METHODS AND ANALYSIS We will search for all randomised controlled trials that compared CBASP, pharmacotherapy or their combination, in the treatment of patients with persistent depressive disorder, in Cochrane CENTRAL, PUBMED, SCOPUS and PsycINFO, supplemented by personal contacts. Individual participant data will be sought from the principal investigators of all the identified trials. Our primary outcomes are depression severity as measured on a continuous observer-rated scale for depression, and dropouts for any reason as a proxy measure of overall treatment acceptability. We will conduct a one-step IPD-NMA to compare CBASP, medications and their combinations, and also carry out a meta-regression to identify their prognostic factors and effect moderators. The model will be fitted in OpenBUGS, using vague priors for all location parameters. For the heterogeneity we will use a half-normal prior on the SD. ETHICS AND DISSEMINATION This study requires no ethical approval. We will publish the findings in a peer-reviewed journal. The study results will contribute to more finely differentiated therapeutics for patients suffering from this chronically disabling disorder. TRIAL REGISTRATION NUMBER CRD42016035886.