A comparison of analytic series methods for Laplacian free boundary problems
A general free boundary problem (Neumann conditions on the free section of the boundary, Dirichlet and Neumann conditions on known sections of it) has been chosen as a sufficiently complicated pattern for problems relative to the (plane) Laplace equation. Approximating the free boundary by splines reduces the problem to solving a sequence of problems with known boundaries. One of the most widely applied analytical techniques to solve problems of this kind (providing accurate, computationally efficient solutions) is the use of series expansions, based on the method of separation of variables. Three methods to estimate the series coefficients (from matrix equations, all involving inner products, i.e. integrals) are proposed: quasi-least squares, least squares, and eigenfunction expansions), and their efficiency (measured by the computational amount) is discussed. Taking all factors into account, the eigenfunction expansion (found e.g. in the treatment of seepage due to the author in [Math. Comput. Modelling 17, No. 12, 9-19 (1993; Zbl 0783.31001), and 18, No. 7, 107 (1993; Zbl 0798.31002)]\ appears to be most suitable in this context.
scientific article; zbMATH DE number 746256