Summary
Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH 1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH 1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL p space then follow.
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This research was performed while the second author was visiting professor at the University of Delaware, spring 1989
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Eggermont, P.P.B., Saranen, J. L p estimates of boundary integral equations for some nonlinear boundary value problems. Numer. Math. 58, 465–478 (1990). https://doi.org/10.1007/BF01385636
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DOI: https://doi.org/10.1007/BF01385636