George C. Hsiao

A finite element method for some integral equations of the first kind

Abstract This paper discusses a finite element approximation for a class of singular integral equations of the first kind. These integral equations are deduced from Dirichlet problems for strongly elliptic differential equations in two independent variables. By a variation of technique due to Aubin, it is shown that the Galerkin method with finite elements as trial functions leads to an optimal rate of convergence.

Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics

The problem of error estimation in the numerical solution of integral equations that arise in electromagnetics is addressed. The direct method (Greens theorem or field approach) and the indirect method (layer ansatz or source approach) lead to well-known integral equations both of the first kind [electric field integral equations (EFIE)] and the second kind [magnetic field integral equations (MFIE)]. These equations are analyzed systematically in terms of the mapping properties of the integral operators. It is shown how the assumption that field quantities have finite energy leads naturally to describing the mapping properties in appropriate Sobolev spaces. These function spaces are demystified through simple examples which also are used to demonstrate the importance of knowing in which space the given data lives and in which space the solution should be sought. It is further shown how the method of moments (or Galerkin method) is formulated in these function spaces and how residual error can be used to estimate actual error in these spaces. The condition number of all of the impedance matrices that result from discretizing the integral equations, including first kind equations, is shown to be bounded when the elements are computed appropriately. Finally, the consequences of carrying out all computations in the space of square integrable functions, a particularly friendly Sobolev space, are explained.

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkins method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.ZusammenfassungWir entwickeln hier ein modifiziertes Kollokationsverfahren als vollständige numerische Diskretisierung des Galerkin-Verfahrens zur Lösung von Fredholmschen Integralgleichungen erster Art mit logarithmischem Kern als Hauptteil. Diese Methode verknüpft hohe Genauigkeit der Galerkin-Verfahren mit der Schnelligkeit von Kolookationsmethoden. Die asymptotische Fehleranalysis liefert optimale Konvergenzordnung im Sinne finiter Element-Approximation. Das Verfahren gehört zu den verbesserten Randelement-Methoden, die zur Lösung einer großen Klasse ebener Randwertprobleme unter Verwendung finiter Elemente auf der Randkurve verwendet werden können. Die numerischen Experimente bestätigen hohe Genauigkeit bei kurzen Rechenzeiten.

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

SummaryIn this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.

Domain decomposition methods via boundary integral equations

Domain decomposition methods are designed to deal with coupled or transmission problems for partial dierential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of dierent discretization schemes. In general, the coupled problem is reduced to the Schur complement equation on the skeleton of the domain decomposition. Boundary integral equations are used to describe the local Steklov{Poincar e operators which are basic for the local Dirichlet{Neumann maps. Using dierent representations of the Steklov{Poincar e operators we formulate and analyze various boundary element methods employed in local discretization schemes. We give sucient conditions for the global stability and derive corresponding a priori error estimates. For the solution of the resulting linear systems we describe appropriate iterative solution strategies using both local and global preconditioning techniques.

Weak Solutions of Fluid–Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gardings inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.

On the Dirichlet problem in elasticity for a domain exterior to an arc

Abstract We consider here a Dirichlet problem for the two-dimensional linear elasticity equations in the domain exterior to an open arc in the plane. It is shown that the problem can be reduced to a system of boundary integral equations with the unknown density function being the jump of stresses across the arc. Existence, uniqueness as well as regularity results for the solution to the boundary integral equations are established in appropriate Sobolev spaces. In particular, asymptotic expansions concerning the singular behavior for the solution near the tips of the arc are obtained. By adding special singular elements to the regular splines as test and trial functions, an augmented Galerkin procedure is used for the corresponding boundary integral equations to obtain a quasi-optimal rate of convergence for the approximate solutions.

A Dynamic Explanation of The Hybrid Effect

A theoretical analysis has been performed to evaluate the dynamic stress concentration factor in hybrid composites composed of low elongation (LE) and high elongation (HE) fibers. At a fracture of a fiber, the present model predicts two stress waves propagating along each fiber in the hybrid. The phase difference of the dynamic responses contributed by these waves at the middle section of a fiber immediately adjacent to a fiber breakage is controll ed by the fiber mass per unit length. The magnitude of the dynamic responses is determined by the fiber extensional stiffness. The study concludes that hybrid effect always exists in a beneficial manner and the case of the parent LE fiber composite provides the upper bound for the dynamic stress concen tration factor in hybrids. The validity of the approximations made in the derivations has been examined.

On the stability of integral equations of the first kind with logarithmic kernels

This paper discusses the stability of the Galerkin method for a class of boundary integral equations of the first kind. These integral equations arise in acoustics, elasticity, and hydrodynamics, and the kernels of the principal parts of the corresponding integral operators all have logarithmic singularities. It is shown that an optimal choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence of the approximate solutions. The results here are consistent with those obtained by the Tikhonov regularization procedure.

On the Two-Dimensional Exterior Boundary-Value Problems of Elasticity

The method of the regularized integral equation is applied to the second fundamental boundary-value problems of elasticity. Both static and dynamic problems are considered. It is shown that in either case the displacement vector can be expressed as a Neumann series in terms of the prescribed stress on the internal boundary.