A comparison of analytic series methods for Laplacian free boundary problems

Solutions to Laplace's equation are required for a wide range of problems. Arguably, the most difficult class of problems involves a “free” boundary, where the location of one (or more) of the boundaries is initially unknown. Analytical solutions for these problems were restricted to regular boundary geometries. However, recently the classical series method has been modified, to cater for arbitrary boundary geometries, using least squares methods. For free boundary problems, solutions can be obtained by solving a sequence of known boundary problems—at each iteration, the series coefficients can be estimated. Efficient calculation of the series coefficients becomes very important, particularly when the number of iterations is relatively high. In this paper, three methods for estimating the series coefficients will be described, in the context of a free boundary problem. The computational cost of each method will be analysed and compared, and the most appropriate method for this class of problem is indicated.     

Non-reflecting boundary conditions for the compressible Euler and Navier-Stokes equations in the finite volume method

The paper is concerned with the numerical implementation of the inlet and outlet boundary conditions in the finite volume method for the solution of the 3D Euler and Navier-Stokes equations. The explicit time marching procedure is described. The classical Riemann problem is modified for physically relevant boundary conditions with the aim to keep conservation laws. This technique was used in [2]. The initial condition in the Riemann problem is replaced by the suitable one-sided boundary condition. This results in the acceleration of the numerical method itself. On the inlet the pressure and the density and the angle of attack or velocity vector and the entropy are prescribed. On the outlet the pressure or normal component of the velocity or temperature or mass flow are investigated in such a way to obtain the unique solution of the modified Riemann problem. Various combinations of inlet and outlet boundary conditions are investigated. This results in the sufficiently precise approximation of real flow boundary conditions. Numerical examples illustrating the usefulness of the proposed approach for cascade flow are presented. Another numerical example is shown in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A symmetric boundary element method for the Stokes problem in multiple connected domains

We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.     

Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition

A priori error estimates in the H¹- and L²-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L²-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

Wavelet solutions for the Dirichlet problem

Summary. A modified classical penalty method for solving a Dirichlet boundary value problem is presented. This new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains. It is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral. The compact support and orthonormality of the basis are essential for representing the boundary measure numerically, and therefore for implementing this methodology. Received June 3, 1992 / Revised version received November 8, 1993     

On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners

The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.     

On Non-Linear Ordinary Differential Equations of Boundary Layer Type

A numerical method previously applied to linear two-point boundary value problems of boundary layer type is extended to some non-linear problems. Discretization of the differential equation leads to a set of non-linear algebraic equations, which is solved by a modified Newton's method; both the mesh spacing and the boundary layer parameter are iteratively adjusted during the solution process. Several examples are discussed; one of these concerns the problem of shock wave formation in a supersonic nozzle.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

Solution Blow-up in a Nonlinear System of Equations with Positive Energy in Field Theory

A problem for a nonlinear system of electromagnetic equations in the Coulomb calibration with allowance for sources of free-charge currents is considered. The local-in-time solvability in the weak sense of the corresponding initial–boundary value problem is proved by applying the method of a priori estimates in conjunction with the Galerkin method. A modified Levine method is used to prove that, for an arbitrary positive initial energy, under a certain initial condition on the functional \(\Phi (t) = \int\limits_\Omega {|A{|^2}dx} \), where A(x) is a vector potential, the solution of the initial–boundary value problem blows up in finite time. An upper bound for the blow-up time is obtained.     

二阶非线性椭圆型方程带位移的边值问题

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain. For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincare boundary value problem.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

B-spline solution of non-linear singular boundary value problems arising in physiology

We use B-spline functions to develop a numerical method for computing approximations to the solution of non-linear singular boundary value problems associated with physiology science. The original differential equation is modified at singular point then the boundary value problem is treated by using B-spline approximation. The numerical method is tested for its efficiency by considering three model problems from physiology.     

An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large.     

Solution blowup for the heat equation with double nonlinearity

We consider a model initial boundary value problem for the heat equation with double nonlinearity. We use a modified Levin method to prove the solution blowup.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

Initial boundary value problem for modified Zakharov equations

In this paper the authors consider the existence and uniqueness of the solution to the initial boundary value problem for a class of modified Zakharov equations, prove the global existence of the solution to the problem by a priori integral estimates and Galerkin method.     

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.     

The Treatment of Boundary Singularities in Axially Symmetric Problems Containing Discs

Axially symmetric problems (e.g. Laplace's equation in cylindricalcoordinates) containing discs possess boundary singularitiesarising from the mixed boundary conditions that occur acrossthe disc edge. A modified finite-difference method is presentedwhich effectively eliminates the inaccuracies that occur inthe standard numerical solution near such singularities. Techniquesfor developing the analytical forms of such singularities aregiven and modified finite-difference approximations are obtained.The steady-state diffusion of oxygen around a circular electrodeis taken as the model problem and a modified quadrature methodis presented for the calculation of the oxygen flux throughthe electrode.