On the convergence of the collocation method for nonlinear boundary integral equations (original) (raw)
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.
On the boundary element method for some nonlinear boundary value problems
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.
Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations
Applied Numerical Mathematics, 2020
Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev-Gauss-Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equation. The proposed method enables us to derive optimal error estimates in L 2 -norm for the Legendre collocation under the two kinds of Lipschitz conditions, respectively. Our method is also applied to the numerical solutions of some nonlinear partial differential equations by using the Legendre Galerkin and Chebyshev collocation in spatial discretization. Numerical examples are given to show the efficiency of the methods.
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.
Collocation Method for Nonlinear Volterra-Fredholm Integral Equations
Open Journal of Applied Sciences, 2012
A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of Volterra-Fredholm integral equations.
High-order collocation methods for nonlinear delay integral equation
Journal of Computational and Applied Mathematics, 2017
The classical collocation methods based on piecewise polynomials have been studied for delay Volterra integral equations of the second-kind in [5]. These collocation methods have uniform order m for any choice of the collocation parameters and can achieve local superconvergence in the grid points by choosing the suitable collocation parameters. In this paper with the aim of increasing the order of classical collocation methods, we use a general class of multistep methods based on Hermite collocation methods and prove that this numerical method has uniform order 2m + 2r for r previous time steps and m collocation points. Some numerical examples are given to show the validity of the presented method and to confirm our theoretical results.
Convergence of meshfree collocation methods for fully nonlinear parabolic equations
Numerische Mathematik
We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.
Numerical solution of problems with non-linear boundary conditions
Mathematics and Computers in Simulation, 2003
In this paper, we are concerned with an elliptic problem in a bounded two-dimensional domain equipped with a non-linear Newton boundary condition. This problem appears, e.g. in the modelling of electrolysis procedures. We assume that the non-linearity has a polynomial behaviour. The problem is discretized by the finite element (FE) method with conforming piecewise linear or polynomial approximations. This problem has been investigated in [Num. Math. 78 (1998) 403; Num. Funct. Anal. Optimiz. 20 (1999) 835] in the case of a polygonal domain, where the convergence and error estimates are established. In [Feistauer et al., On the Finite Element Analysis of Problems with Non-linear Newton Boundary Conditions in Non-polygonal Domains, in press] the convergence of the FE approximations to the exact solution is proved in the case of a nonpolygonal domain with curved boundary. The analysis of the error estimates leads to interesting results. The non-linearity in boundary condition causes the decreas of the approximation error. Further decreas is caused by the application of the numerical integration in the computation of boundary integrals containing the non-linear terms. In [Feistauer et al., Numerical analysis of problems with non-linear Newton boundary conditions, in: Proceedings of the Third Conference of ENUMATH'99, p. 486], numerical experiments prove that this decreas is not the result of a poor analysis, but it really appears. In our paper, we give a brief of the results. The main attention is paid to the development of the error estimates for higher-order FE method. The error estimates are compared with experiments.
Convergent collocation methods for parabolic equations
arXiv: Numerical Analysis, 2018
We show that the functions constructed by the collocation methods with Wendland kernels converge to unique viscosity solutions of the corresponding fully nonlinear parabolic equations. The key ingredients in our proof are the stability property of Wendland kernels that states the norms of the interpolation operators are bounded provided that the inverses of the interpolation matrices exhibit good decays, and the max-min representations of the nonlinearities of the equations. Several numerical experiments support our assumption and result.
A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel
Journal of Computational and Applied Mathematics, 2010
This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.
The Discrete Galerkin method for nonlinear integral equations
Journal of Integral Equations and Applications, 1988
Let K be a completely continuous nonlinear integral operator, and consider solving x = K(x) by Galerkin's method. This can be written as x n = PnK(x n ),Pn an orthogonal projection; the iterated Galerkin solution is defined by x n = K(x n ). We give a general framework and error analysis for the numerical method that results from replacing all integrals in Galerkin's method with numerical integrals. A special high order formula is given for integral equations arising from solving nonlinear two-point boundary value problems.