On the boundary element method for some nonlinear boundary value problems

Ruotsalainen, K.; Wendland, Wolfgang L.

doi:10.1007/bf01404466

Cited by 54 publications

(38 citation statements)

References 33 publications

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“…We give complete error analysis of some practically computable approximations to the solution of (1.1) and (1.2). The convergence rates obtained in this work are similar to those obtained in [18] and in the sequel work. But, earlier results cover only the two-dimensional case while results in our work are for two-and three-space problems with Lipschitz boundaries.…”

Section: Introductionsupporting

confidence: 86%

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Boundary element methods for potential problems with nonlinear boundary conditions

Ganesh

Steinbach

2000

Math. Comp.

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Abstract. Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two-and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

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Section: Introductionsupporting

confidence: 86%

“…Earlier work in solving (1.4) was based on a direct boundary integral formulation initiated in [18] and further studied in [1,7,9,10,17]. In the direct formulation the nonlinear operator appears as density of the double layer potential, adding additional difficulty in dealing with the nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Boundary element methods for potential problems with nonlinear boundary conditions

Ganesh

Steinbach

2000

Math. Comp.

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“…In the case when the body force f = f (x) is independent of u, based on the theory of monotone operators, this problem has been treated in [11] via boundary integral equation methods and was motivated by a similar problem for the Laplacian in [17], [16]. For the nonlinear body force f (u), in principle, by following [4], we may imbed both (5.1) and (5.7) into a family of nonlinear Robin problems.…”

Section: Discussionmentioning

confidence: 99%

A Newton-imbedding procedure for solutions of semilinear boundary value problems in Sobolev spaces

Hsiao

2006

Complex Variables and Elliptic Equations

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This paper is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of the method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H 1 (Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity will also be discussed.AMS: 35J20, 35J60, 45B05, 65N30. KEY WORDS: nonlinear problem, Newton-imbedding iteration, variational formulation, boundary integral equations, a priori estimate.

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“…If it is 1, then (1.1)-(1.2) can be redefined on a rescaled region D in such a way that the new C Γ = 1 (see [5]). The solvability of (1.4) follows from the results of [10]. We introduce a parametrization r(t) = (ξ(t), η(t)), 0 ≤ t ≤ 2π, for the boundary Γ.…”

Section: Introductionmentioning

confidence: 99%

An extrapolation method for a class of boundary integral equations

Zhao

1996

Math. Comp.

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Abstract. Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

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On the boundary element method for some nonlinear boundary value problems

Cited by 54 publications

References 33 publications

Boundary element methods for potential problems with nonlinear boundary conditions

Boundary element methods for potential problems with nonlinear boundary conditions

A Newton-imbedding procedure for solutions of semilinear boundary value problems in Sobolev spaces

An extrapolation method for a class of boundary integral equations

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