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Boundary-field equation methods for a class of nonlinear problems. (English) Zbl 0832.65126

Pitman Research Notes in Mathematics Series. 331. Harlow: Longman House. New York, NY: Wiley, 178 p. (1995).
The book consists of the following chapters: 1. The coupling of the boundary element method (BEM) and the finite element method (FEM) for linear problems, 2. Monotone nonlinear problems, 3. Asymptotic error analysis, 4. An elasto-plastic problem, 5. Nonmonotone nonlinear problems, 6. The uncoupling of BEM and FEM for nonlinear problems.
Chapter 1 begins with a brief explanation of the idea of coupling: Consider e.g. the exterior Dirichlet problem for the Poisson equation: given a bounded domain \(\Omega_0\) with smooth boundary \(\Gamma_0\), find a \(u\) such that (1) \(-\Delta u = f\) in \(\Omega^C_0\), \(u = g\) on \(\Gamma_0\) and \(u = O(|x|^{-1})\) as \(|x|\to \infty\), where \(f\), \(g\) are given smooth functions. By means of an auxiliary surface \(\Gamma\) we divide \(\Omega^C_0\) into two subdomains: the bounded domain \(\Omega^-\), whose boundary consists of two parts \(\Gamma_0\), \(\Gamma\) and the unbounded domain \(\Omega^+\). We suppose that \(\Omega^-\) contains the support of \(f\). The problem (1) is now replaced by a transmission problem: (2) \(-\Delta u = f\) in \(\Omega^-\), \(u = g\) on \(\Gamma_0\), (3) \(-\Delta u = 0\) in \(\Omega^+\), \(u = O(|x|^{-1})\) as \(x \to \infty\), (4) \({\partial u^- \over \partial \nu} = {\partial u^+ \over \partial \nu}\), \(u^- = u^+\) on \(\Gamma\) (the sign \({}^-\) resp. \({}^+\) denotes inner resp. outer limit on \(\Gamma\) of the corresponding function). Then (2) is solved by the finite element method and problem (3) is reduced, using (4), to the integral equation on \(\Gamma\), which may be obtained in various ways. This method is applied in Chapter 1 to linear problems as the exterior Dirichlet problem in \(\mathbb{R}^3\) and the Stokes problem in \(\mathbb{R}^2\).
In Chapter 2 the nonlinear exterior boundary value problem is considered, the operator \(\Delta u\) being replaced by one of the form (5) \(\sum_i {\partial \over \partial x_i} a_i (x, \nabla u(x))\). After obtaining the weak formulation of the problem its equivalence is shown with an operator equation of the form (6) \(T(w, \sigma) = F\), where the unknonw functions \(w\), \(\sigma\) are defined in \(\Omega^-\) and on \(\Gamma\), respectively, and \(T\) is acting in the product of some spaces of Sobolev type. This equation may be treated using the theory of monotone operators. Some known theorems concerning such operators are gathered and used in the sequel for proving the existence and uniqueness of the solution. In the last section the Galerkin approximations of the exact problem are studied and the unique solvability of the Galerkin equations together with error estimates are proved.
Chapter 3 contains a detailed error analysis of the finite element approximations of equation (6). It is supposed that \(T\) is strongly monotone and Lipschitz continuous.
In Chapter 4 the above presented methods are applied for solving a three- dimensional elasto-plastic interface problem, particularly the unique solvability of the exact problem and an error estimate for the Galerkin approximations are proved.
Chapter 5 deals with a slightly more general operator than (5), namely \(- \sum_i {\partial \over \partial x_i} a_i (\cdot, u, \nabla u)+a_0(\cdot,u, \nabla u)\) giving two different coupling procedures. The operator \(T\) occurring in the corresponding equation of type (6) in this case is not a monotone one, only bounded and continuous. The solvability of (6) (which is equivalent to a weak formulation of the considered exterior boundary value problem) is proved using a sequence of Galerkin approximations.
Chapter 6 is devoted to a modification of the coupling method treated in Chapters 1-5. The main idea of this method (called the uncoupling) is as follows: the auxiliary surface \(\Gamma\) considered in the coupling method is now a circle in \(\mathbb{R}^2\) or a sphere in \(\mathbb{R}^3\). This allows to invert exactly the boundary integral operators appearing in the coupling equations. As a result the weak formulation of the problem becomes much simpler and at the discrete level the work of coding and computation is strongly reduced. The method of uncoupling is first explained in the case of linear equation (1) and then applied to the nonlinear problems previously considered in Chapters 2 and 5.
The book contains detailed references concerning the treated problems.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
74J99 Waves in solid mechanics
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