Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Summary The standard 5-point difference scheme for the model problem u=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.     

The composite mini element: a new mixed FEM for the Stokes equations on complicated domains

We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two and three-dimensional domains that may contain a huge number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. It turns out that the method can be viewed as a coarse scale generalization of the classical mini element approach, i.e. it reduces the computational effort while the approximation quality depends linearly on the (coarse) mesh size in the usual way. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Parameter-Uniform B-Spline Collocation Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems

We consider a Dirichlet boundary value problem for a class of singularly perturbed semilinear reaction-diffusion equations. A  B-spline collocation method on a piecewise-uniform Shishkin mesh is developed to solve such problems numerically. The convergence analysis is given and the method is shown to be almost second-order convergent, uniformly with respect to the perturbation parameter ε in the maximum norm. Numerical results are presented to validate the theoretical results.     

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.     

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.     

Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

Problem of steady motion for a second-grade fluid in the hölder classes of functions

The paper studies a boundary-value problem (with the usual adherence boundary condition) for a stationary system of equations of motion of second-grade fluids in a bounded domain. This system is not elliptic and contains third-order derivatives of the velocity vector field. This leads to obvious difficulties in the analysis of the problem. It is known that the problem is reduced to the usual Stokes problem and to the transport equations or their analogs. We present a new easier method of such a reduction which allows us to prove the solvability of a stationary boundary-value problem for the equations of motion of second-grade fluids in the Hölder classes of functions in the case of small exterior forces. Bibliography: 6 titles.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

样条有限元

本文用三次B样条变分方法解规则区域上板梁组合弹性结构的平衡问题.推导出了适用于各种边界条件的统一计算格式,便于在计算机上实现.与通常有限元相比,具有计算量少、精确度高等显著特点.文中对自然边界条件作为约束条件的影响给予了考虑,并以板的弯曲问题为例说明影响极微.给出了几个数值的例子.     

A preconditioning strategy for boundary element Galerkin methods

The Dirichlet and Neumann problems for the Laplacian are reformulated in the usual way as boundary integral equations of the first kind with symmetric kernels. These integral equations are solved using Galerkin's method with piecewise-constant and piecewise-linear boundary elements, respectively. In both cases, the stiffness matrix is symmetric and positive-definite, and has a condition number of order N, the number of degrees of freedom. By contrast, the condition number of the product of the two stiffness matrices is bounded independently of N. Hence, we can use the Neumann stiffness matrix to precondition the Dirichlet stiffness matrix, and vice versa. © 1997 John Wiley & Sons, Inc.     

Homogenization of a fluid problem with a free boundary

The stationary Stokes equations with a free boundary are studied in a perforated domain. The perforation consists of a periodic array of cylinders of size and distance O(ε). The free boundary is given as the graph of a function on a two‐dimensional perforated domain. We derive equations for the two‐scale limit of solutions. The limiting equation is a free boundary system. It involves a nonlinear eliptic operator corresponding to the nonlinear mean‐curvature expression in the original equations. Depending on the equation for the contact angle, the pressure is in general unbounded. © 2000 John Wiley & Sons, Inc.     

Necessary conditions for distributed-parameter systems with controls of fewer variables than state variables

Necessary conditions are proved for multi-dimensional control systems, in which the state is a function of several independent variables in a fixed domain, while the controlu is a function of only some of the independent variables (that is,u is a lower-dimensional control). The cost functional can be taken in the usual Lagrange form of a multiple integral or in Mayer form. The state equations are a system of partial differential equations in explicit form with assigned boundary conditions, and constraints may be imposed on the values of the control variables.The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper and Professor L. DeBoer for his interest and encouragement. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.     

A Time Finite Element Method Based on the Generalized Variational Calculus

In this contribution, a new finite element method in the temporal domain is presented, in which the time step size is introduced as an additional variable. Thus, the variation of the time integral of the Lagrangean resulting from Hamilton's principle has to be carried out with respect to the rules of the generalized variational calculus. Apart from the usual time integral of the Euler‐Lagrange differential equations, the so‐called transversality condition is obtained as an additional result representing a time‐boundary term, which is used to obtain an optimal step size in the time domain.     

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.     

A numerical method for problems in infinite strips with irregularities extending to infinity

The Dirichlet-to-Neumann (DtN) Finite Element Method is a combined analytic-numerical method for boundary value problems in infinite domains. The use of this method is usually based on the assumption that, in the infinite domain D exterior to the finite computational domain, the governing differential equations are sufficiently simple. In particular, in D it is generally assumed that the equations are linear, homogeneous, and have constant coefficients. In this article, an extension of the DtN method is proposed, which can be applied to elliptic problems with “irregularities” in the exterior domain D, such as (a) inhomogeneities, (b) variable coefficients, and (c) nonlinearities. This method is based on iterative “regularization” of the problem in D, and on the efficient treatment of infinite-domain integrals. Semi-infinite strip problems are used for illustrating the method. Convergence of the iterative process is analyzed both theoretically and numerically. Nonuniformity difficulties and a way to overcome them are discussed. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:233–249, 1998     

A modified scaled boundary finite element method for dynamic response of a discontinuous layered half-space

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.