Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

Summary In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

On the fast solutions of nonlinear elliptic equations

Summary In a recent paper we described a multi-grid algorithm for the numerical solution of Fredholm's integral equation of the second kind. This multi-grid iteration of the second kind has important applications to elliptic boundary value problems. Here we study the treatment of nonlinear boundary value problems. The required amount of computational work is proportional to the work needed for a sequence of linear equations. No derivatives are required since these linear problems are not the linearized equations.     

On the boundary element method for some nonlinear boundary value problems

Summary Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Neas.Using properties of the linearised equations, we can also prove quasioptimal convergence of the spline Galerkin approximations.This work was carried out while the first author was visiting the University of Stuttgart     

The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with non-constant coefficients and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with constant coefficients and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birthThis work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme Boundary Element Methods within the guest-programme We-659/19-2.     

Spectral approximation of the periodic-nonperiodic Navier-Stokes equations

Summary In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an infsup condition. We analyze a spectral and a collocation pseudo-spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave likeM ^– whereM depends on the number of degrees of freedom of the method and represents the regularity of the data.     

Spline collocation for singular integro-differential equations over (0.1)

Summary This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane

Summary In this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions. It turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed+1. Some numerical examples which conform the theoretical results are presented. Present address: (1. 7. 1988–31. 12. 1988) Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Summary A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL ^p error estimates are derived for 2p. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.     

L 2 estimates for the finite element method for the Dirichlet problem with singular data

Summary In this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous Dirichlet condition on the boundary. In these problems the data are Borel measures. Using a quasiuniform mesh of finite elements and polynomials of degree 1, we prove that in two dimensions the convergence is of orderh inL ² and in three dimensions of orderh ^1/2.     

Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners

Summary The eigenvalue problem of the Laplace operator is considered on a non-convex domain composed of rectangles. This model problem may be solved by the finite element method with bilinear elements on a rectangular mesh. It is known thatO(h) ²(<1) convergence can be obtained for the eigenvalues, if the mesh hasO(h) ^–2 points. A simple extrapolation scheme is presented which, on appropriately graded meshes, increases the rate of convergence toO(h) ⁴ This work was supported by the Deutsche Forschungsgemeinschaft (DFG), SFB 123 Stochatistische Mathematische Modelle, Universität Heidelberg     

An error expansion for cubature with an integrand with homogeneous boundary singularities

Summary A common strategy in the numerical integration over ann-dimensional hypercube or simplex, is to consider a regular subdivision of the integration domain intom ⁿ subdomains and to approximate the integral over each subdomain by means of a cubature formula. An asymptotic error expansion whenm is derived in case of an integrand with homogeneous boundary singularities. The error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains.     

The finite element method with nodes moving along the characteristics for convection-diffusion equations

Summary We examine the convergence properties of the finite element method with nodes moving along the characteristics for one-dimensional convection-diffusion equations. For linear elements, we demonstrate optimal rates of convergence in theL ²,H ¹ andL norms. Both linear and nonlinear problems are considered.This work forms part of the research programme of the Oxford/Reading Institute for Computational Dynamics.     

A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow     

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries

Summary. We consider an indirect boundary integral equation formulation for the mixed Dirichlet-Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space. Received April 15, 1995 / Revised version received April 10, 1996     

Convergence rate estimate for a domain decomposition method

Summary We provide a convergence rate analysis for a variant of the domain decomposition method introduced by Gropp and Keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with Dirichlet boundary conditions in ². We show that the convergence rate of the preconditioned GMRES method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures, if the global coarse mesh is fine enough.This author was supported by the National Science Foundation under contract numbers DCR-8521451 and ECS-8957475, by the IBM Corporation, and by the 3M Company, while in residence at Yale UniversityThis author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under Contract W-31-109-Eng-38This author was supported by the National Science Foundation under contract number ECS-8957475, by the IBM Corporation, and by the 3M Company