Discrete Fourier analysis of multigrid algorithms

The main topic of this report is a detailed discussion of the discrete Fourier multilevel analysis of multigrid algorithms. First, a brief overview of multigrid methods is given for discretizations of both linear and nonlinear partial differential equations. Special attention is given to the hp-Multigrid as Smoother algorithm, which is a new algorithm suitable for higher order accurate discontinuous Galerkin discretizations of advection dominated flows. In order to analyze the performance of the multigrid algorithms the error transformation operator for several linear multigrid algorithms are derived. The operator norm and spectral radius of the multigrid error transformation are then computed using discrete Fourier analysis. First, the main operations in the discrete Fourier analysis are defined, including the aliasing of modes. Next, the Fourier symbol of the multigrid operators is computed and used to obtain the Fourier symbol of the multigrid error transformation operator. In the multilevel analysis, two and three level h-multigrid, both for uniformly and semi-coarsened meshes, are considered, and also the analysis of the hp-Multigrid as Smoother algorithm for three polynomial levels and three uniformly and semi-coarsened meshes. The report concludes with a discussion of the multigrid operator norm and spectral radius. In the appendix some useful auxiliary results are summarized

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained

A Hybridizable Discontinuous Galerkin Method for the Navier–Stokes Equations with Pointwise Divergence-Free Velocity Field

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation

Three-field block preconditioners for models of coupled magma/mantle dynamics

For a prescribed porosity, the coupled magma/mantle flow equations can be formulated as a two-field system of equations with velocity and pressure as unknowns. Previous work has shown that while optimal preconditioners for the two-field formulation can be obtained, the construction of preconditioners that are uniform with respect to model parameters is difficult. This limits the applicability of two-field preconditioners in certain regimes of practical interest. We address this issue by reformulating the governing equations as a three-field problem, which removes a term that was problematic in the two-field formulation in favour of an additional equation for a pressure-like field. For the three-field problem, we develop and analyse new preconditioners and we show numerically that they are optimal in terms of problem size and less sensitive to model parameters, compared to the two-field preconditioner. This extends the applicability of optimal preconditioners for coupled mantle/magma dynamics into parameter regimes of physical interest

On fixed-point, Krylov, and 2×22\times 2 block preconditioners for nonsymmetric problems

The solution of matrices with $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, this paper proves a direct equivalence between convergence of $2\times2$ block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of $2\times 2$ block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. Theoretical results are verified numerically on a nonsymmetric steady linearized Navier-Stokes discretization, which also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses.Comment: Accepted to SIMA

Analysis of a hybridized/interface stabilized finite element method for the stokes equations

Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise solenoidal. It is shown that the method is inf-sup stable for both equal-order and locally Taylor--Hood type spaces, and \emph{a priori} error estimates are developed. The considered method can be constructed to have the same global algebraic structure as a conforming Galerkin method, unlike standard discontinuous Galerkin methods that have greater number of degrees of freedom than conforming Galerkin methods on a given mesh. We assert that this method is amongst the simplest and most flexible finite element approaches for Stokes flow that provide local mass conservation. With this contribution the mathematical basis is established, and this supports the performance of the method that has been observed experimentally in other works

A prospective study into change of vitamin D levels, depression and frailty among depressed older persons

Objectives While vitamin D is involved in frailty as well as depression, hardly any study has examined the course of vitamin D levels prospectively. The objective of this study is to examine whether a change of vitamin D in depressed older adults is associated with either depression course, course of frailty, or both. Methods The study population consisted of 232 of 378 older adults (60-93 years) with a DSM-IV defined depressive disorder participating in the Netherlands Study of Depression in Older persons, a prospective clinical cohort study. Baseline and 2-year follow-up data on depressive disorder (DSM-IV diagnosis), symptom severity (inventory of depressive symptoms), frailty phenotype (and its individual components) and vitamin D levels were obtained. Linear mixed models were used to study the association of change in vitamin D levels with depression course, course of frailty, and the combination. Results Vitamin D levels decreased from baseline to follow-up, independent from depression course. An increase in frailty was associated with a significantly sharper decrease of vitamin D levels over time. Post hoc analyses showed that this association with frailty might be driven by an increase of exhaustion over time and counteracted by an increase in walking speed. Conclusions Our findings generate the hypothesis that vitamin D supplementation in late-life depression may improve frailty, which may partly explain inconsistent findings of randomised controlled trials evaluating the effect of vitamin D for depression. We advocate to consider frailty (components) as an outcome in future supplementation trials in late-life depression

Frailty measures in immuno-metabolic subtypes of late-life depression; A two-year prospective study:A two-year prospective study

Background/Objectives - Frailty is highly prevalent with increasing age. Based on the concept of depression as a disorder of accelerated aging and its association with inflammation and metabolic dysregulation, we examined whether frailty measures at baseline and over time differed between immuno-metabolic subtypes of late-life depression. Methods - Clinical cohort study in primary and secondary mental health care with two-year follow-up. In total 359 depressed older patients (≥ 60 years) classified in four immuno-metabolic subgroups by latent profile analysis. We compared frailty measures at baseline and two-year follow-up adjusted for confounders between immuno-metabolic based depressed subgroups. Frailty measures included the frailty index, physical frailty phenotype, and two proxies (handgrip strength, gait speed). Results - At baseline, the relatively healthy depressed subgroup (n = 181) performed best on all frailty markers. While frailty markers worsened over time, the two-year course did not differ between the subgroups for any of these markers. Conclusion - The more severe immuno-metabolic dysregulation present in late-life depression, the more frail. Nonetheless, as trajectories over time did not differ between subgroups, the difference probably emerged at midlife. Future studies should examine whether geriatric assessment might become relevant at earlier ages in specialized mental health care

Yield gap analysis and entry points for improving productivity on large oil palm plantations and smallholder farms in Ghana

Oil palm production must increase in Ghana to meet the increasing demand for palm oil and avoid costly imports. Although maximum fruit bunch (FB) yields of >20 t ha−1 yr−1 are achievable, average FB yields in Ghana are only 7 t ha−1 yr−1. Despite the pressing need to increase palm oil production and improve yields, knowledge of the underlying causes of poor yields in Ghana is lacking. Closing yield gaps in existing plantings in smallholdings and plantations offers great opportunities to increase oil production without area expansion, thus sparing land for other uses. This study sought to understand the magnitude and underlying causes of yield gaps in plantation and smallholder oil palm production systems in Ghana based on a detailed characterization of management practices and yield measurements over a two-year period. Using a boundary line analysis, the water-limited yield (Yw) over a planting cycle was defined as about 21 t ha−1 yr−1 FB, with yield gaps of 15.4 t ha−1 yr−1 FB at smallholder farms and 9.8 t ha−1 yr−1 FB at plantations. Poor management practices, including incomplete crop recovery (i.e., harvesting all suitable crop) and inadequate agronomic management were the main factors contributing to these yield gaps. Productivity losses were further exacerbated by low oil extraction rates by small-scale processors of 12% as compared to 21% by the large-scale processors. The potential losses in annual crude palm oil (CPO) during the crop plateau yield phase therefore exceed 5 and 3 t ha−1 yr−1 for small-scale and large-scale production systems respectively. Investment to reduce yield gaps by appropriate agronomic and yield recovery practices across all production systems, while improving access of smallholder producers to more efficient oil palm processing facilities, can make a significant contribution to closing the supply gap for palm oil in Ghana. The impact of such investments on large-scale plantations could result in a doubling of CPO production. Smallholder farmers could benefit the most with a fourteen-fold increase in CPO production and economic gains of >1 billion US$