Computing the Newton Potential in the Boundary Integral Equation for the Dirichlet Problem of the Poisson Equation

Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier-Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has mth-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order (n-m log3 n), with requiring (n log2 n) number of arithmetics for the computation, where n is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier-Garlerkin method

Proceedings of the 12th UK Conference on Boundary Integral Methods (UKBIM12)

Boundary integral methods have become established for solving a wide variety of problems in science and engineering. UK based researchers have been active and made substantial contributions in the theory and development of boundary integral formulations, as well as their analysis, discretisation and numerical solution. The UKBIM conference series aims to provide a forum where recent developments in boundary integral methods can be discussed in an informal atmosphere. The first UK conference on boundary integral methods (UKBIM) was held at the University of Leeds in 1997. Subsequent UKBIM conferences have taken place in Brunel (1999), Brighton (2001), Salford (2003), Liverpool (2005), Durham (2007), Nottingham (2009), Leeds (2011), Aberdeen (2013), Brighton (2015) and Nottingham-Trent (2017). The success of these events has made the conference a regular event for researchers based in the UK, and elsewhere, who are working on all aspects of boundary integral methods. This book contains the abstracts and papers presented at the Twelfth UK Conference on Boundary Integral Methods (UKBIM 12), held at Oxford Brookes University in July 2019. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield. I am grateful to the members of the scientific committee for their advice and support during the past year, and to all the authors and reviewers for their hard work in producing the high quality peer-reviewed papers for this book

Robust multigrid methods for Isogeometric discretizations applied to poroelasticity problems

El análisis isogeométrico (IGA) elimina la barrera existente entre elementos finitos (FEA) y el diseño geométrico asistido por ordenador (CAD). Debido a esto, IGA es un método novedoso que está recibiendo una creciente atención en la literatura y recientemente se ha convertido en tendencia. Muchos esfuerzos están siendo puestos en el diseño de solvers eficientes y robustos para este tipo de discretizaciones. Dada la optimalidad de los métodos multimalla para elementos finitos, la aplicación de estosmétodos a discretizaciones isogeométricas no ha pasado desapercibida. Nosotros pensamos firmemente que los métodos multimalla son unos candidatos muy prometedores a ser solvers eficientes y robustos para IGA y por lo tanto en esta tesis apostamos por su aplicación. Para contar con un análisis teórico para el diseño de nuestros métodos multimalla, el análisis local de Fourier es propuesto como principal análisis cuantitativo. En esta tesis, a parte de considerar varios problemas escalares, prestamos especial atención al problema de poroelasticidad, concretamente al modelo cuasiestático de Biot para el proceso de consolidación del suelo. Actualmente, el diseño de métodos multimalla robustos para problemas poroelásticos respecto a parámetros físicos o el tamaño de la malla es un gran reto. Por ello, la principal contribución de esta tesis es la propuesta de métodos multimalla robustos para discretizaciones isogeométricas aplicadas al problema de poroelasticidad.La primera parte de esta tesis se centra en la construcción paramétrica de curvas y superficies dado que estas técnicas son la base de IGA. Así, la definición de los polinomios de Bernstein y curvas de Bézier se presenta como punto de partida. Después, introducimos los llamados B-splines y B-splines racionales no uniformes (NURBS) puesto que éstas serán las funciones base consideradas en nuestro estudio.La segunda parte trata sobre el análisis isogeométrico propiamente dicho. En esta parte, el método isoparamétrico es explicado al lector y se presenta el análisis isogeométrico de algunos problemas. Además, introducimos la formulación fuerte y débil de los problemas anteriores mediante el método de Galerkin y los espacios de aproximación isogeométricos. El siguiente punto de esta tesis se centra en los métodos multimalla. Se tratan las bases de los métodos multimalla y, además de introducir algunos métodos iterativos clásicos como suavizadores, también se introducen suavizadores por bloques como los métodos de Schwarz multiplicativos y aditivos. Llegados a esta parte, nos centramos en el LFA para el diseño de métodos multimalla robustos y eficientes. Además, se explican en detalle el análisis estándar y el análisis basado en ventanas junto al análisis de suavizadores por bloques y el análisis para sistemas de ecuaciones en derivadas parciales.Tras introducir las discretizaciones isogeométricas, los métodos multimalla y el LFA como análisis teórico, nuestro propósito es diseñar métodos multimalla eficientes y robustos respecto al grado polinomial de los splines para discretizaciones isogeométricas de algunos problemas escalares. Así, mostramos que el uso de métodos multimalla basados en suavizadores de tipo Schwarz multiplicativo o aditivo produce buenos resultados y factores de convergencia asintóticos robustos. La última parte de esta tesis está dedicada al análisis isogeométrico del problema de poroelasticidad. Para esta tarea, se introducen el modelo de Biot y su discretización isogeométrica. Además, presentamos una novedosa estabilización de masa para la formulación de dos campos de las ecuaciones de Biot que elimina todas las oscilaciones no físicas en la aproximación numérica de la presión. Después, nos centramos en dos tipos de solvers para estas ecuaciones poroelásticas: Solvers desacoplados y solvers monolíticos. En el primer grupo, le dedicamos una especial atención al método fixed-stress y a un método iterativo propuesto por nosotros que puede ser aplicado de forma automática a partir de la estabilización de masa ya mencionada.Por otro lado, realizamos un análisis de von Neumann para este método iterativo aplicado al problema de Terzaghi y demostramos su estabilidad y convergencia para los pares de elementos Q1 Q1, Q2 Q1 y Q3 Q2 (con suavidad global C1). Respecto al grupo de solvers monolíticos, nosotros proponemos métodos multimalla basados en suavizadores acoplados y desacoplados. En esta parte, métodosIsogeometric analysis (IGA) eliminates the gap between finite element analysis (FEA) and computer aided design (CAD). Due to this, IGA is an innovative approach that is receiving an increasing attention in the literature and it has recently become a trending topic. Many research efforts are being devoted to the design of efficient and robust solvers for this type of discretization. Given the optimality of multigrid methods for FEA, the application of these methods to IGA discretizations has not been unnoticed. We firmly think that they are a very promising approach as efficient and robust solvers for IGA and therefore in this thesis we are concerned about their application. In order to give a theoretical support to the design of multigrid solvers, local Fourier analysis (LFA) is proposed as the main quantitative analysis. Although different scalar problems are also considered along this thesis, we make a special focus on poroelasticity problems. More concretely, we focus on the quasi-static Biot's equations for the soil consolidation process. Nowadays, it is a very challenging task to achieve robust multigrid solvers for poroelasticity problems with respect physical parameters and/or the mesh size. Thus, the main contribution of this thesis is to propose robust multigrid methods for isogeometric discretizations applied to poroelasticity problems. The first part of this thesis is devoted to the introduction of the parametric construction of curves and surfaces since these techniques are the basis of IGA. Hence, with the definition of Bernstein polynomials and B\'ezier curves as a starting point, we introduce B-splines and non-uniform rational B-splines (NURBS) since these will be the basis functions considered for our numerical experiments. The second part deals with the isogeometric analysis. In this part, the isoparametric approach is explained to the reader and the isogeometric analysis of some scalar problems is presented. Hence, the strong and weak formulations by means of Galerkin's method are introduced and the isogeometric approximation spaces as well. The next point of this thesis consists of multigrid methods. The basics of multigrid methods are explained and, besides the presentation of some classical iterative methods as smoothers, block-wise smoothers such as multiplicative and additive Schwarz methods are also introduced. At this point, we introduce LFA for the design of efficient and robust multigrid methods. Furthermore, both standard and infinite subgrids local Fourier analysis are explained in detail together with the analysis for block-wise smoothers and the analysis for systems of partial differential equations. After the introduction of isogeometric discretizations, multigrid methods as our choice of solvers and LFA as theoretical analysis, our goal is to design efficient and robust multigrid methods with respect to the spline degree for IGA discretizations of some scalar problems. Hence, we show that the use of multigrid methods based on multiplicative or additive Schwarz methods provide a good performance and robust asymptotic convergence rates. The last part of this thesis is devoted to the isogeometric analysis of poroelasticity. For this task, Biot's model and its isogeometric discretization are introduced. Moreover, we present an innovative mass stabilization of the two-field formulation of Biot's equations that eliminates all the spurious oscillations in the numerical approximation of the pressure. Then, we deal with two types of solvers for these poroelastic equations: Decoupled and monolithic solvers. In the first group we devote special attention to the fixed-stress split method and a mass stabilized iterative scheme proposed by us that can be automatically applied from the mass stabilization formulation mentioned before. In addition, we perform a von Neumann analysis for this iterative decoupled solver applied to Terzaghi's problem and demonstrate that it is stable and convergent for pairs Q1-Q1, Q2-Q1 and Q3-Q2 (with global smoothness C1). Regarding the group of monolithic solvers, we propose multigrid methods based on coupled and decoupled smoothers. Coupled additive Schwarz methods are proposed as coupled smoothers for isogeometric Taylor-Hood elements. More concretely, we propose a 51-point additive Schwarz method for the pair Q2-Q1. In the last part, we also propose to use an inexact version of the fixed-stress split algorithm as decoupled smoother by applying iterations of different additive Schwarz methods for each variable. For the latter approach, we consider the pairs of elements Q2-Q1 and Q3-Q2 (with global smoothness C1). Finally, thanks to LFA we manage to design efficient and robust multigrid solvers for the Biot's equations and some numerical results are shown.<br /

Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Des avancées dans la réduction de modèle de type PGD pour les EDPs d’ordre élevé, le traitement des géométries complexes et la résolution des équations de Navier-Stokes instationnaires

The main purpose of this work is to describe a simulation method for the use of aPGD-based Model reduction Method (MOR) for solving high order partial differentialequations. First, the PGD method is used for solving fourth order PDEs and thealgorithm is illustrated on a lid-driven cavity problem. Transformations of coordinatesfor changing the complex physical domain into the simple computational domain arealso studied, which lead to extend the spatial PGD method to complex geometrydomains. Some numerical examples for different kinds of domain are treated toillustrate the potentialities of this methodology.Finally, a PGD-based space-time separation is introduced to solve the unsteadyStokes or Navier-Stokes equations. This decomposition makes use of common tem-poral modes for both velocity and pressure, which lead to velocity spatial modessatisfying individually the incompressibility condition. The adaptation and imple-mentation of a PGD approach into a general purpose finite volume framework isdescribed and illustrated on several analytic and academic flow examples. A largereduction of the computational cost is observed on most of the treated examples.L’objectif principal de ce travail est de proposer une nouvelle approche de simulationbasée sur une Méthode de réduction du modèle (MOR) utilisant une décompositionPGD. Dans ce travail, cette approche est d’abord utilisée pour résoudre des équationsaux dérivées partielles d’ordre élevé avec un exemple numérique pour les équations auxdérivées partielles du quatrième ordre sur le problème de la cavité entraînée. Ensuiteun changement de coordonnées pour transformer le domaine physique complexe enun domaine de calcul simple est étudié, ce qui conduit à étendre la méthode PGDau traitement de certaines géométries complexes. Divers exemples numériques pourdifférents types de domaines géométriques sont ainsi traités avec l’approche PGD.Enfin, une séparation espace-temps est proposée pour résoudre les équations deNavier-Stokes instationnaires à l’aide d’une approche PGD. Cette décompositionest basée sur le choix de modes temporels communs pour la vitesse et la pression,ce qui conduit à une décomposition basée sur des modes spatiaux satisfaisant in-dividuellement la condition d’incompressibilité. L’adaptation d’une formulationvolumes finis à cette décomposition PGD est présentée et validée sur de premiersexemples analytiques ou académiques pour les équations de Stokes ou Navier-Stokesinstationnaires. Une importante réduction des temps calculs est observée sur lespremiers exemples traités

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds