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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Courant, R., Hilbert, D.: Methods of mathematical physics. N.Y.: Interscience 1962
Elliot, C.H., Ockendon, J.R.: Weak and variational methods for moving boundary problems. Research Notes in Mathematics No. 59. London: Pitman 1982
Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput28, 963–971 (1979)
Fasano, A., Primicerio, M.: Convergence of Huber's method for heat conduction problems with change of phase. Z. Angew. Math. Mech.53, 341–348 (1973)
Fasano, A., Primicerio, M.: Free boundary problems: Theory and applications, Montecatini, Research Notes in Mathematics Nos. 78 and 79. Boston: Pitman 1983
Gidas, B., Li, N.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys.68, 209–243 (1979)
Glimm, J.: Solution in the large for nonlinear hyperbolic systems. Comm. Pure Appl. Math.18, 695–715 (1965)
Kinderlehrer, D., Stampacchia, G.: An Inroduction to variational inequalities and their applications. New York: Academic Press 1980
Magenes, E.: Free boundary problems. Istituto Nazionale di Alta Matematica Francesco Severi, Rome, 1980
Meyer, G.H.: On the computational solution of elliptic and parabolic free boundary problems, in Free Boundary Problems (E. Magenes, ed.), Istituto Nazionale di Alta Matematica Francesco Severi, Rome, 1980
Meyer, G.H.: The method of lines and invariant imbedding for elliptic and parabolic free boundary problems, SIAM J. Number. Anal.18, 150–164 (1981)
Meyer, G.H.: An analysis of the method of lines for the Reynolds Equation in hydrodynamic lubrication, SIAM J. Numer Anal.18, 165–177 (1981)
Ockendon, T., Hodgkins, W.: Moving boundary problems in heat flow and diffusion. Oxford: Clarendon Press 1975
Roberts, G.: Diffusion with chemical reaction. Metal Science. Feb. 1979, pp. 94–97
Sage, A.: Optimum systems control. Englewood Cliffs: Prentice-Hall 1968
Schultz, M.N.: Spline analysis. Englewood Cliffs: Prentice-Hall 1973
Wilson, D.G., Solomon, A.D., Boggs, P.T.: Moving boundary problems. New York: Academic Press 1978
Author information
Authors and Affiliations
Additional information
This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145
Rights and permissions
About this article
Cite this article
Meyer, G.H. Free boundary problems with nonlinear source terms. Numer. Math. 43, 463–483 (1984). https://doi.org/10.1007/BF01390185
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01390185