Dirichlet Domain Decomposition Methods

by Yi Zhang

Coupling interface data is THE most important component of domain decomposition (DD) methods. In many cases the coupled data are only required to up to order 1, which means, only continuity of variable and continuity of normal derivatives are need for adequate accuracy.  So, in variant formulation of the transmission condition of a problem, when its domain is decomposed, one can require a priori that the Dirichlet boundary/interface condition are built in the trial function, while the Neumann condition at boundary and interface has to be enforced explicitly. In particular, with u_i(x) as unknown variable in the local subdomain \Omega_i and t_{i}(x) as its exterior normal derivative at the subdomain boundary \Gamma_i, the Dirichlet transmission condition

u_i=u_j

for neighboring subdomains \Omega_i and \Omega_j at the common “border” \Gamma_{ij}, can be built in the trial function.  The Neumann condition at the same interface would be

t_i(x)+t_j(x)=0

because the normal derivatives are to opposite direction. At this moment, I should make some definitions. Let’s call the “real” boundary of the domain as “boundary”, while the local “boundary” for certain subdomain, together with “real” boundary, as skeleton \Gamma_S. So condition above is imposed on skeletons. We should also consider some Neumann boundary condition t=\nabla u\cdot n=g_N is imposed at part of the boundary \Gamma_N, and Dirichlet boundary condition u=g_D is imposed on the rest, called \Gamma_D. Now the weak form of N. boundary/interface is

\int_{\Gamma_{ij}}{(t_i+t_j)v}=0

\int_{\Gamma_i\cap\Gamma_N}{(t_i-g_N)v}=0

For one interface/boundary of one subdomain, only one of equations above is valid. When those equations are added up for all the subdomains, we have

\sum\int_{\Gamma_i}{t_iv}ds=\int_{\Gamma_N}{g_Nv}ds

If we look those normal derivatives as flux, and notice that the summation does not include the boundaries in \Gamma_D, above equation simply says that the total flux across the skeletons and N. boundaries adds up to the flux across N. boundaries. In correspondence, the test function v(x) would be zero on \Gamma_D. Besides, with the notation above, the interfaces in the inner of domain is \Gamma_S-\Gamma_N\cap\Gamma_D.

Dirichlet DD method,  resolves u(x), by which the Dirichlet boundary condition and interface matching condition are naturally satisfied. The remaining the problem now is then how to obtain a weak formulation with unknown u from the last equation above. One can make an educated guess, if there exists a map between u_i(x) and t_i(x), say, t_i=\mathcal{F}(u_i), we immediately have the weak formulation of u(x) as

\sum\int_{\Gamma_i}{\mathcal{F}(u_i)v}ds_x=\int_{\Gamma_N}{g_Nv}ds_x

It turns out that that map \mathcal{F} does exist, under some condition of course. And by noticing that the we did not employ any information of the specific differential equation, one can guess that those information are hidden exactly in \mathcal{F}.

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