View source for Schwarz alternating method

One of the general methods for finding a solution to the [[Dirichlet problem|Dirichlet problem]]; it allows one to obtain a solution to the Dirichlet problem for a differential equation of elliptic type in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834901.png" /> that can be represented as the union of a finite number of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834902.png" /> in which the solution to the Dirichlet problem is already known. Studies of H.A. Schwarz (1869; see [[#References|[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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834904.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834906.png" /> are discs, then the solution to the Dirichlet problem for each of them is given by the [[Poisson integral|Poisson integral]]. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834907.png" /> be the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s0834909.png" /> for which a solution to the Dirichlet problem is sought (see Fig.). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349010.png" /> denote the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349011.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349012.png" /> denote the parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349013.png" /> that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349014.png" /> (they are interior in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349015.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349016.png" /> be the remaining parts, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349017.png" />. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349018.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349020.png" /> are its parts that fall in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349021.png" /> (they are also interior in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349022.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349023.png" /> are the remaining parts, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349024.png" />. Then the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349026.png" /> can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349027.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083490a.gif" />

Figure: s083490a

Now, given a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349029.png" />, one has to find a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349031.png" /> that is continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349032.png" /> and that takes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349033.png" /> the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349034.png" />. The restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349036.png" /> can be continuously extended to the whole boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349037.png" />, and for these boundary values one finds a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349038.png" /> to the Dirichlet problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349039.png" />. The values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349041.png" /> together with the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349043.png" /> now form a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349044.png" /> for which a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349045.png" /> to the Dirichlet problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349046.png" /> is found. Further, a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349047.png" /> to the Dirichlet problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349048.png" /> is constructed, based on the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349052.png" />, etc. The sought function has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349053.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349054.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349055.png" />.

A method analogous to the Schwarz alternating method (see [[#References|[2]]]) can be applied to finding a solution to the Dirichlet problem in the intersection of two domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349057.png" />, if its solutions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083490/s08349059.png" /> 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 [[#References|[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 [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.A. Schwarz, "Gesamm. math. Abhandl." , '''2''' , Springer (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Neumann, ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl.'' , '''22''' (1870) pp. 264–321</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.35301}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Nevanlinna, "Uniformisierung" , Springer (1967) {{MR|0228671}} {{ZBL|0152.27401}} </TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.L. Helms, "Introduction to potential theory" , Wiley (Interscience) (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T.F. Chan, et al., "Domain decomposition methods for partial differential equations" , SIAM (1990) {{MR|1064333}} {{ZBL|0695.00026}} </TD></TR></table>