The boundary elements method for magneto-hydrodynamic (MHD) channel flows at high Hartmann numbers

In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 10⁵${10}^5$ by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.     

A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers

This article aims to study the partitioned method for magnetohydrodynamic flows at small magnetic Reynolds numbers. We design a partitioned second‐order method and show that this method is stable under a time step () restrict condition. Our method can decouple the magnetohydrodynamic equations so that we can solve two relatively simple subproblems separately at each time step, which is computationally economic. A complete theoretical analysis of error estimates is also given. Finally, we present numerical experiments to support our theory.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1966–1986, 2017     

A boundary only procedure for time-dependent diffusion equations

In the recent literature, the boundary element method (BEM) is extensively used to solve time-dependent partial differential equations. However, most of these formulations yield algorithms where one has to include all interior points in the computation process if finite difference procedures are used to approximate the temporal derivative. This obviously restricts the advantages of the BEM, which is mainly considered to be a boundary only algorithm for time-independent problems. A new algorithm is demonstrated here, which extends the boundary only nature of the method to time-dependent partial differential equations. Using this procedure, one can reduce the finite difference time integration algorithm, generated in a standard manner, to a boundary only process. The proposed method is demonstrated with considerable success for diffusion problems. Results obtained in these applications are presented comparatively with analytical and other boundary element time integration procedures. The algorithm proposed may utilize several coordinate functions in the secondary reduction phase of the formulation. A summary of such functions is described here and performances of these functions are tested and compared in three applications. It is shown that some coordinate functions perform better than others under certain conditions. Using these results, we propose a general coordinate function, which may be used with satisfactory results in all parabolic partial differential equation applications.     

A new defect correction method for the Navier-Stokes equations at high Reynolds numbers

In this paper, a new defect correction method for the Navier-Stokes equations is presented. With solving an artificial viscosity stabilized nonlinear problem in the defect step, and correcting the residual by linearized equations in the correction step for a few steps, this combination is particularly efficient for the Navier-Stokes equations at high Reynolds numbers. In both the defect and correction steps, we use the Oseen iterative scheme to solve the discrete nonlinear equations. Furthermore, the stability and convergence of this new method are deduced, which are better than that of the classical ones. Finally, some numerical experiments are performed to verify the theoretical predictions and show the efficiency of the new combination.     

A boundary element method for homogenization of periodic structures

Homogenized coefficients of periodic structures are calculated via an auxiliary partial differential equation in the periodic cell. Typically, a volume finite element discretization is employed for the numerical solution. In this paper, we reformulate the problem as a boundary integral equation using Steklov–Poincaré operators. The resulting boundary element method only discretizes the boundary of the periodic cell and the interface between the materials within the cell. We prove that the homogenized coefficients converge super-linearly with the mesh size, and we support the theory with examples in two and three dimensions.     

A cell boundary element method for elliptic problems

An elementary analysis on the cell boundary element (CBEM) was given by Jeon and Sheen. In this article we improve the previous results in various aspects. First of all, stability and convergence analysis on the rectangular grids are established. Moreover, error estimates are improved. Our improved analysis was possible by recasting of the CBEM in a Petrov‐Galerkin setting. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined‐field integral equation formulations for time‐harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the L² condition numbers for these formulations and also on the norms of the classical acoustic single‐ and double‐layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number k, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like k^1/3 as k →∞, when the scatterer is a circle or sphere, it can grow as fast as k^7/5 for a class of “trapping” obstacles. In this article, we prove further bounds, sharpening and extending our previous results. In particular, we show that there exist trapping obstacles for which the condition numbers grow as fast as exp(γk), for some γ > 0, as k →∞ through some sequence. This result depends on exponential localization bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low k. In the second part of the article, we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A BOUNDARY ELEMENT METHOD FOR A NONLINEAR BOUNDARY VALUE PROBLEM IN STEADY-STATE HEAT TRANSFER IN DIMENSION THREE

In this paper, a new method of boundary reduction is proposed, which reduces thesteady-state heat transfer equation with radiation. Moreover, a boundary element method is pre-sented for its solution and the error estimates of the numerical approximations are given.     

Defect correction method for viscoelastic fluid flows at high Weissenberg number

We study a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step. We prove the convergence of the defect correction method and derive an error estimate for the Oseen‐viscoelastic model problem. The derived theoretical results are supported by numerical tests for both the Oseen‐viscoelastic problem and the Johnson‐Segalman model problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A numerical solution for advection‐diffusion equation using dual reciprocity method

This article describes a numerical method based on the boundary integral equation and dual reciprocity method(DRM) for solving the one‐dimensional advection‐diffusion equations. The concept of DRM is used to convert the domain integral to the boundary that leads to an integration free method. The time derivative is approximated by the time‐stepping method. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of the new approach. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A convolution quadrature Galerkin boundary element method for the exterior Neumann problem of the wave equation

The numerical solution of the Neumann problem of the wave equation on unbounded three‐dimensional domains is calculated using the convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The mathematical analysis that has been built up for the Dirichlet problem is extended and developed for the Neumann problem, which is important for many modelling applications. Numerical examples are then presented for one of these applications, modelling transient acoustic radiation. Copyright © 2009 John Wiley & Sons, Ltd.     

A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.     

The boundary element method with Lagrangian multipliers

On open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi‐optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment. The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The solution of the non-homogeneous Helmholtz equation by means of the boundary element method

One of the most important advantages of the Boundary Element Method (BEM) is that no internal discretization of the domain is required. This advantage, however, is generally lost when source terms are present in the governing differential equation. It is shown here that for the non-homogeneous Helmholtz equation with a harmonic source term, it is possible to transform the volume integral into a surface integral thus retaining this feature. The transformation is achieved using the Green formula. The technique is applied to solve numerically a test problem with known simple analytical solution.     

A new family of high regularity elements

In this article, we propose a new family of high regularity finite element spaces. The global approximation spaces are obtained in two steps. We first build an open cover of the computational domain and local approximation spaces on each patch of the cover. Then we construct partition of unity functions subordinate to the open cover depending on the regularity requirement. The basis functions of the global space is given by the products of the local basis functions and the corresponding partition of unity functions. The method can be used to construct finite element spaces of any desired regularity. Approximation properties and implementation details are discussed. Numerical examples for the biharmonic equation are presented to show the effectiveness of the proposed method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 1–16, 2012