Summary
The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.
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This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145
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Meyer, G.H. Free boundary problems with nonlinear source terms. Numer. Math. 43, 463–483 (1984). https://doi.org/10.1007/BF01390185
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DOI: https://doi.org/10.1007/BF01390185