On the convergence of the collocation method for nonlinear boundary integral equations (original) (raw)
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Journal of Integral Equations and Applications, 1992
Recently, Galerkin and collocation methods have been analyzed in connection with the nonlinear boundary integral equation which arises in solving the potential problem with a nonlinear boundary condition. Considering this model equation, we propose here a discretized scheme such that the nonlinearity is replaced by its L 2 -orthogonal projection. We are able to show that this approximate collocation scheme preserves the theoretical L 2 -convergence. For piecewise linear continuous splines, our numerical experiments confirm the theoretical quadratic L 2 -convergence.
On the collocation method for a nonlinear boundary integral equation
Journal of Computational and Applied Mathematics, 1989
In this paper we study a potential problem with a nonlinear boundary condition. Using the Green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation. We shall give a brief discussion of the solvability of the integral equation. The aim of this paper, however, is to analyse the collocation method for finding an approximate solution to this equation. Using the theory of a-proper and a-stable mappings we prove the unique solvability of the collocation equations and the asymptotic error'estimates. To do this we assume that the nonlinearity is strongly monotone.
Some applications of a galerkin‐collocation method for boundary integral equations of the first kind
Mathematical Methods in the Applied Sciences, 1984
Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the fist kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order-1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using E-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in Lz. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptoticd agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles. *) This research was supported by the " Alexander-von-Humboldt-Stiftung". the "Deutsche Forschungsgemeinschaft" (WE 659). the "Applied Mathematics Institute" and the
Faedo–Galerkin’s method for a non-linear boundary value problem
We consider a nonlinear boundary value problem. The use of Faedo–Galerkin techniques and a compactness result, while passing to the limit, we prove the existence of the variational solution of the considered problem. A new result is given by showing the uniqueness of the solution basing on the hypotheses which are weaker than those considered by J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris, Dunod, (1969; (1969; Zbl 0189.40603)] for a similar problem. We will finish by studying the regularity of the obtained solution.
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Numerische Mathematik, 1988
Here we analyse the boundary element Galerkin method for twodimensional 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 Ne~as. Using properties of the linearised equations, we can also prove quasioptimal convergence of the spline Galerkin approximations.
Applied Numerical Mathematics, 1995
This paper is concerned with the proof of convergence and derivation of convergence rates of a perturbed polynomial collocation method for the numerical solution of Cauchy-type singular integral equations. The analysis is restricted to the constant coefficients and nonnegative index case. The integrals involved in the compact operator are approximated by quadrature rules.
The numerical solution of a non-linear boundary integral equation on smooth surfaces
IMA Journal of Numerical Analysis, 1994
We study a boundary integral equation method for solving Laplace's equation u = 0 with nonlinear boundary conditions. This nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the collocation method, with piecewise quadratic functions used as approximations to u. Convergence results are given for the cases that (1) the original surface is used, and (2) the surface is approximated by piecewise quadratic interpolation. In addition, we de ne and analyze a two-grid iteration method for solving the nonlinear system that arises from the discretization of the boundary integral equation. Numerical examples are given and the paper concludes with a short discussion of the relative cost of di erent parts of the method.
Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method
Numerische Mathematik, 1972
Let u be the solution of the differential equation L u (x) =/(x, u (x)) for xe(0,1) (with appropriate boundary conditions), where L is an elliptic differential operator. Let ~ be the Galerkin approximation to u with polynomial spline trial functions. We obtain error bounds of the form HDi(u-~!}]Lp<=Chk-,i][Dku]~Lp, where 0-<_]~m andm<k~2m+q,p=2orp=oo, histhe 'mesh size' and qis a non negative integer depending on the splines being used.
The Galerkin and Collocation Methods for Two-Dimensional Integral Equations
2018
The Galerkin and Collocation methods are regularization strategies for inverse problems. They are based on projection operators, which are formal strategies to discretize the equation. In ”An Introduction to Mathematical Theory of Inverse Problems”, written by Andreas Kirsch, the two methods are applied for unidimensional integral equations, i.e., with only one independent variable. But there are problems in which the integral equation is bidimensional. This paper presents the Galerkin and Collocation Methods for integral equations with two independent variables.