Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations

We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator

Flux evaluation in primal and dual boundary-coupled problems

A crucial aspect in boundary-coupled problems such as fluid-structure interaction pertains to the evaluation of fluxes. In boundary-coupled problems, the flux evaluation appears implicitly in the formulation and, consequently, improper flux evaluation can lead to instability. Finite-element approximations of primal and dual problems corresponding to improper formulations can therefore be non-convergent or display suboptimal convergence rates. In this paper, we consider the main aspects of flux evaluation in finite-element approximations of boundary-coupled problems. Based on a model problem, we consider various formulations and illustrate the implications for corresponding primal and dual problems. In addition, we discuss the extension to free-boundary problems, fluid-structure interaction, and electro-osmosis applications

A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations

While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model

A unified framework for navier-stokes cahn-hilliard models with non-matching densities

Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of Navier-Stokes Cahn-Hilliard models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible Navier-Stokes Cahn-Hilliard models, (3) variations between the models only appear in the constitutive choices. The framework presented in this work now completes the fundamental exploration of alternate non-matching density Navier-Stokes Cahn-Hilliard models that utilize a single momentum equation for the mixture velocity, but leaves open room for further sophistication in the energy functional and constitutive dependence.Comment: Corrections; 49 page

Categorizing Different Approaches to the Cosmological Constant Problem

We have found that proposals addressing the old cosmological constant problem come in various categories. The aim of this paper is to identify as many different, credible mechanisms as possible and to provide them with a code for future reference. We find that they all can be classified into five different schemes of which we indicate the advantages and drawbacks. Besides, we add a new approach based on a symmetry principle mapping real to imaginary spacetime.Comment: updated version, accepted for publicatio

Pharmacogenomic associations of adverse drug reactions in asthma: systematic review and research prioritisation

A systematic review of pharmacogenomic studies capturing adverse drug reactions (ADRs) related to asthma medications was undertaken, and a survey of Pharmacogenomics in Childhood Asthma (PiCA) consortia members was conducted. Studies were eligible if genetic polymorphisms were compared with suspected ADR(s) in a patient with asthma, as either a primary or secondary outcome. Five studies met the inclusion criteria. The ADRs and polymorphisms identified were change in lung function tests (rs1042713), adrenal suppression (rs591118), and decreased bone mineral density (rs6461639) and accretion (rs9896933, rs2074439). Two of these polymorphisms were replicated within the paper, but none had external replication. Priorities from PiCA consortia members (representing 15 institution in eight countries) for future studies were tachycardia (SABA/LABA), adrenal suppression/crisis and growth suppression (corticosteroids), sleep/behaviour disturbances (leukotriene receptor antagonists), and nausea and vomiting (theophylline). Future pharmacogenomic studies in asthma should collect relevant ADR data as well as markers of efficacy

Goal-Adaptive Discretization of Fluid-Structure Interaction

The simulation of complex physical phenomena, such as fluid–structure interaction, appears to be within reach in view of the significant progress in computing power over the last decades. Yet, we are still far away from what is desirable in an evermore-demanding, science-and-technology-based society. If, however, we are modest with what we need of a physical system in that we ask for specific goal quantities instead of the entire solution, then we are able to save tremendous amounts of computing effort.Mechanics, Aerospace Structures & MaterialsAerospace Engineerin

An H1(Ph)-Coercive Discontinuous Galerkin Formulation for The Poisson Problem: 1-D Analysis

Discontinuous Galerkin (DG) methods are finite element techniques for the solution of partial differential equations. They allow shape functions which are discontinuous across inter-element edges. In principle, DG methods are ideally suited for hp-adaptivity, as they handle nonconforming meshes and varying-in-space polynomial-degree approximations with ease. Recently, DG formulations for elliptic problems have been put in a general framework of analysis. Although clarifying basic properties, the analysis does not warrant a clear preference. Specifically, none of the conventional DG formulations possesses a bilinear form that is coercive (and continuous) on an infinite-dimensional broken Sobolev space. Rather, bilinear forms are only weakly coercive or defined on subspaces only and employ stabilization parameters that typically increase unboundedly as the subspace is expanded, e.g., if the polynomial degree is increased. For hp-adaptation, coercivity is a fundamental property: By the classical Lax-Milgram theorem, any conforming discretization of a coercive formulation is stable, i.e., discrete approximations are well-posed and have a unique solution, irrespective of the specifics of the underlying approximation space. In this thesis we consider the one-dimensional Poisson problem and present a generic consistent conventional DG formulation. We show that conventional DG formulations are necessarily noncoercive. Moreover, we presents a new symmetric DG formulation which contains nonconventional edge terms based on element Green's functions and the data local to the edges. We show that the new DG formulation is coercive on H1(Ph), the space of functions that are piecewise in the H1 Sobolev space. Furthermore, we show that the new DG formulation and the classical Galerkin formulation are equivalent, that is, in the infinite-dimensional case they yield the same solution.Aerospace Engineerin

Isogeometric analysis-based goal-oriented error estimation for free-boundary problems

We consider goal-oriented error estimation for free-boundary problems using isogeometric analysis. Goal-oriented methods require the solution of the dual problem, which is a problem for the adjoint of the linearized free-boundary problem. Owing to linearization, this dual problem includes a curvature-dependent boundary condition, which leads to cumbersome implementations if the discrete free boundary is only continuous, as in a piecewise-linear representation. Isogeometric finite elements straightforwardly provide continuously differentiable free boundaries for which the corresponding dual problem can be easily implemented. We illustrate the computation of the linearized-adjoint problems with two test cases and estimate the error in corresponding quantities of interest. In the first problem, a single B-spline patch can be employed. In the second problem, we employ T-splines. Bézier extraction is used to provide a finite element interface to these two distinct spline technologies