Schwarz alternating method

One of the general methods for finding a solution to the Dirichlet problem; it allows one to obtain a solution to the Dirichlet problem for a differential equation of elliptic type in domains that can be represented as the union of a finite number of domains in which the solution to the Dirichlet problem is already known. Studies of H.A. Schwarz (1869; see [1]) and a number of later studies by other authors were dedicated to this method for finding a solution to the Dirichlet problem for the Laplace equation in plane domains. The principal idea of the Schwarz alternating method as applied to the simplest case of the Laplace equation in the union of two plane domains is the following.

Let and be two domains in the plane with non-empty intersection and such that the solution to the Dirichlet problem for the Laplace equation is known for each of them. For instance, if and are discs, then the solution to the Dirichlet problem for each of them is given by the Poisson integral. Further, let be the union of and for which a solution to the Dirichlet problem is sought (see Fig.). Let denote the boundary of , let denote the parts of that are in (they are interior in ) and let be the remaining parts, so that . Similarly, is the boundary of , are its parts that fall in (they are also interior in ) and are the remaining parts, that is, . Then the boundary of can be represented in the form .

Figure: s083490a

Now, given a continuous function on , one has to find a harmonic function in that is continuous in the closed domain and that takes on the values of . The restriction of to can be continuously extended to the whole boundary , and for these boundary values one finds a solution to the Dirichlet problem in . The values of on together with the values of on now form a continuous function on for which a solution to the Dirichlet problem in is found. Further, a solution to the Dirichlet problem in is constructed, based on the values of on and on , etc. The sought function has the form

and

Using bounded solutions of the Dirichlet problem with piecewise-continuous boundary data allows one to choose the values zero on the remaining parts of the boundaries without having to worry about the continuous extension of .

A method analogous to the Schwarz alternating method (see [2]) can be applied to finding a solution to the Dirichlet problem in the intersection of two domains and , if its solutions for and are known.

Schwarz' alternating method is also used to solve boundary value problems of a more general nature for general equations of elliptic type (including equations of an order greater than two) under certain additional conditions [3], and also in domains in space.

Schwarz' alternating method is extremely important for the construction of various harmonic functions (with pre-assigned singularities) on Riemann surfaces [4].

References

[1]	H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890)
[2]	C. Neumann, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. , 22 (1870) pp. 264–321
[3]	L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) MR0106537 Zbl 0083.35301
[4]	R. Nevanlinna, "Uniformisierung" , Springer (1967) MR0228671 Zbl 0152.27401

Comments

Recently these ideas are meeting new interest among numerical analysts. They are essentially used to solve boundary value problems on complicated domains. Such domains are decomposed in smaller and simpler ones; therefore such methods are referred to as domain decomposition methods.

See [a1], pp. 200-203, for a more subtle application of the Schwarz alternating method in the study of boundary value problems.

References

[a1]	L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)
[a2]	T.F. Chan, et al., "Domain decomposition methods for partial differential equations" , SIAM (1990) MR1064333 Zbl 0695.00026