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2017 | Buch

Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016

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Über dieses Buch

This volume collects papers associated with lectures that were presented at the BAIL 2016 conference, which was held from 14 to 19 August 2016 at Beijing Computational Science Research Center and Tsinghua University in Beijing, China. It showcases the variety and quality of current research into numerical and asymptotic methods for theoretical and practical problems whose solutions involve layer phenomena.

The BAIL (Boundary And Interior Layers) conferences, held usually in even-numbered years, bring together mathematicians and engineers/physicists whose research involves layer phenomena, with the aim of promoting interaction between

these often-separate disciplines. These layers appear as solutions of singularly perturbed differential equations of various types, and are common in physical problems, most notably in fluid dynamics.

This book is of interest for current researchers from mathematics, engineering and physics whose work involves the accurate app

roximation of solutions of singularly perturbed differential equations; that is, problems whose solutions exhibit boundary and/or interior layers.

Inhaltsverzeichnis

Frontmatter
Error Estimates in Balanced Norms of Finite Element Methods on Layer-Adapted Meshes for Second Order Reaction-Diffusion Problems
Abstract
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H 1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss anisotropic problems, semilinear equations, supercloseness and a combination technique. Moreover, we consider different classes of layer-adapted meshes and sketch the three-dimensional case. Remarks to systems and problems with different layers close the paper.
Hans-G. Roos
Numerical Studies of Higher Order Variational Time Stepping Schemes for Evolutionary Navier-Stokes Equations
Abstract
We present in this paper numerical studies of higher order variational time stepping schemes combined with finite element methods for simulations of the evolutionary Navier-Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin–Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.
Naveed Ahmed, Gunar Matthies
Uniform Convergent Monotone Iterates for Nonlinear Parabolic Reaction-Diffusion Systems
Abstract
This paper deals with a uniform convergent monotone method for solving nonlinear singularly perturbed parabolic reaction-diffusion systems. The uniform convergence on a piecewise uniform mesh is established. Numerical experiments are presented.
Igor Boglaev
Order Reduction and Uniform Convergence of an Alternating Direction Method for Solving 2D Time Dependent Convection-Diffusion Problems
Abstract
In this work we solve efficiently 2D time dependent singularly perturbed problems. The fully discrete numerical scheme is constructed by using a two step discretization process, firstly in space, by using the classical upwind finite difference scheme on a special mesh of Shishkin type, and later on in time by using the fractional implicit Euler method. The method is uniformly convergent with respect to the diffusion parameter having first order in time and almost first order in space. We focus our interest on the analysis of the influence of general Dirichlet boundary conditions in the convergence of the algorithm. We propose a simple modification of the natural evaluations, which avoid the order reduction associated to those natural evaluations. Some numerical tests are shown in order to exhibit, from a practical of point of view, the robustness of the numerical method as well as the influence of the improved boundary conditions.
C. Clavero, J. C. Jorge
Laminar Boundary Layer Flow with DBD Plasma Actuation: A Similarity Equation
Abstract
The framework of self-similar laminar boundary layer flow solutions is extended to include the effect of actuation with body force fields resembling those generated by DBD plasma actuators. The deduction line is similar to previous work investigating the effect of porous wall suction on laminar boundary layers. The starting point of the analysis is a generalised form of the Boundary Layer Partial Differential Equations (BL-PDEs) that includes volume force terms. Actuation force distributions are defined such that the volume force term of the BL-PDE equations conforms to the requirements of similarity. New similarity parameters for the plasma strength and thickness are identified. The procedure yields a general similarity equation which includes the effect of pressure gradients, wall transpiration and DBD plasma actuation. Select numerical solutions of the new similarity equation are presented to develop instinctive understanding and prompt a discussion on the construction of new closure relations for integral boundary layer models.
Gael de Oliveira, Marios Kotsonis, Bas van Oudheusden
On Robust Error Estimation for Singularly Perturbed Fourth-Order Problems
Abstract
Recently, several classes of fourth order singularly perturbed problems were considered and uniform convergence in the associated energy norm as well as in a balanced norm was proved. In this proceedings paper we will extend some results by looking into L -bounds and postprocessing.
Sebastian Franz, Hans-Görg Roos
Singularly Perturbed Initial-Boundary Value Problems with a Pulse in the Initial Condition
Abstract
A singularly perturbed parabolic equation of reaction-diffusion type is examined. Initially the solution approximates a concentrated source, which causes an interior layer to form within the solution for all future times. Combining a classical finite difference operator with a layer-adapted mesh, parameter-uniform convergence is established. Numerical results are presented to illustrate the theoretical error bounds.
José Luis Gracia, Eugene O’Riordan
Numerical Results for Singularly Perturbed Convection-Diffusion Problems on an Annulus
Abstract
Numerical methods for singularly perturbed convection-diffusion problems posed on annular domains are constructed and their performance is examined for a range of small values of the singular perturbation parameter. A standard polar coordinate transformation leads to a transformed elliptic operator containing no mixed second order derivative and the transformed problem is then posed on a rectangular domain. In the radial direction, a piecewise-uniform Shishkin mesh is used. This mesh captures any boundary layer appearing near the outflow boundary. The performance of such a method is examined in the presence or absence of compatibility constraints at characteristic points, which are associated with the reduced problem.
Alan F. Hegarty, Eugene O’Riordan
Numerical Calculation of Aerodynamic Noise Generated from an Aircraft in Low Mach Number Flight
Abstract
The paper describes numerical prediction of aerodynamic noise generated from the aircraft. It focuses on the simulation of turbulent flow around rectified flap on the wing represented in 2D. Simulation of turbulent flow is modeled using the stabilized orthogonal subgrid scale (OSGS) method with dynamical subscales. It is shown how the stabilization method can perform simulation of turbulent flow affecting the prediction of acoustic sources calculated applying Lighthill’s analogy. Acoustic sources are used in inhomogeneous Helmholtz equation to simulate pressure wave propagation in the domain closing the circle of three main steps required for simulating aeroacoustics phenomena. It is shown that OSGS with dynamical subscales gives better representation of the spectrum. Overall, better prediction of energy transfer across large and small eddies provides better allocation and presentation of acoustics sources. These sources change wave propagation of the pressure in acoustic field.
Vladimir Jazarević, Boško Rašuo
On the Discrete Maximum Principle for Algebraic Flux Correction Schemes with Limiters of Upwind Type
Abstract
Algebraic flux correction (AFC) schemes are applied to the numerical solution of scalar steady-state convection-diffusion-reaction equations. A general result on the discrete maximum principle (DMP) is established under a weak assumption on the limiters and used for proving the DMP for a particular limiter of upwind type under an assumption that may hold also on non-Delaunay meshes. Moreover, a simple modification of this limiter is proposed that guarantees the validity of the DMP on arbitrary simplicial meshes. Furthermore, it is shown that AFC schemes do not provide sharp approximations of boundary layers if meshes do not respect the convection direction in an appropriate way.
Petr Knobloch
Energy-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes: Neumann Boundary Conditions
Abstract
Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The case of the Dirichlet boundary conditions was considered in the recent article (Kopteva, Numer. Math., 2017, Published online 2 May 2017. doi:10.1007/s00211-017-0889-3). Now we extend this analysis to also allow boundary conditions of Neumann type.
Natalia Kopteva
A DG Least-Squares Finite Element Method for Nagumo’s Nerve Equation with Fast Reaction: A Numerical Study
Abstract
The Nagumo equation is a simple nonlinear reaction-diffusion equation, which has important applications in neuroscience and biological electricity. If the equation is reaction-dominated, numerical oscillations may appear near the traveling wave front, which makes it challenging to find stable solutions. In the present study, a new method is developed on uniform meshes to solve the Nagumo equation. Numerical results are given to demonstrate the performance of the algorithm. Convergence rates with respect to spatial and temporal discretization are obtained experimentally. Some properties of the nerve model are confirmed numerically.
Runchang Lin
Local Projection Stabilization for Convection-Diffusion-Reaction Equations on Surfaces
Abstract
The numerical solution of convection-diffusion-reaction equations in two and three dimensional domains Ω is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known about convection-diffusion-reaction equations on surfaces Γ = ∂Ω. We propose a Local Projection Stabilization (LPS) for convection-diffusion-reaction equations on surfaces based on a linear surface approximation and first order finite elements. Unique solvability of the continuous and discrete problem are established. Numerical test examples show the potential of the proposed method.
Kristin Simon, Lutz Tobiska
A Comparison Study of Parabolic Monge-Ampère Equations Adaptive Grid Methods
Abstract
We consider two recently developed adaptive grid methods for solving time dependent partial differential equations (PDEs) in higher dimensions. These methods compute the adaptive grid based on solving an optimal mass transport problem also known as Monge-Kantorovich problem (MKP). The optimal solution of the MKP is reduced to solving Monge-Ampère equation and is known to have some nice theoretical properties that are desirable for the mesh adaptation. However, these two adaptive grid methods solve the Monge-Ampère equation differently and they are distinctly different in their approaches for computing the adaptive mesh over time. A comparison study to address these various distinctions between the two methods is presented. Several numerical experiments are conducted to illustrate the main differences between the two methods in terms of their mesh quality and performances.
Mohamed H. M. Sulman
Approximate Solutions to Poisson Equation Using Least Squares Support Vector Machines
Abstract
This article deals with Poisson Equations with Dirichlet boundary conditions. A new approach based on least squares support vector machines (LS-SVM) is proposed for obtaining their approximate solutions. The approximate solution is presented in closed form by means of LS-SVM, whose parameters are adjusted to minimize an appropriate error function. The approximate solutions consist of two parts. The first part is a known function that satisfies boundary conditions. The other is two terms product. One term is known function which is zero on boundary, another term is unknown which is related to kernel functions. This method has been successfully tested on rectangle and disc domain and has yielded higher accuracy solutions.
Ziku Wu, Zhenbin Liu, Fule Li, Jiaju Yu
Backmatter
Metadaten
Titel
Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016
herausgegeben von
Prof. Zhongyi Huang
Prof. Martin Stynes
Prof. Zhimin Zhang
Copyright-Jahr
2017
Electronic ISBN
978-3-319-67202-1
Print ISBN
978-3-319-67201-4
DOI
https://doi.org/10.1007/978-3-319-67202-1