Steklov-Poincaré operator for overlapping domain decomposition

Question

For non-overlapping domain decomposition methods for elliptic problems there is an associated Steklov-Poincaré definite positive operator defined on the interface, allowing a direct computation of the interface solution from the boundary data. I have never heard about an equivalent kind of operator for overlapping methods, though I expect it is probably possible that it exists.

Can anyone give me some pointers on materials about that ?

My ambition is to have a unified view between overlapping and non-overlapping methods.

Thank you so much. Tom

Answer 1

The classification of domain decomposition methods into non-overlapping and overlapping methods can be refined.

• The non-overlapping methods have two groups: single trace and multiple trace methods, where `trace' is sometimes the primal variables (aka Dirichlet traces) and sometimes the dual variables (aka Lagrange multipliers) on the interfaces. In the single-trace methods, the adjacent subdomains share a single trace along their common intersections (multiple dual variables are needed at lower dimensional intersections like cross-points in 2-D). In the multiple-trace methods, each subdomain never equips the same trace with neighbouring subdomains but owns a trace for itself. In the non-overlapping case, multiple traces must recover the matching conditions: ui = uj, D ui = D uj for second-order PDEs. For example, just use Dirichlet and Neumann as in the above conditions; or use equivalent linear or non-linear transform of them.

• The overlapping subdomains have to own a trace for itself and can not share the trace with others because the overlapping subdomains even do not share an interface as their common boundaries. So the overlapping methods are also, in this sense, multiple-trace methods. We need only one matching condition on each of the two interfaces $\Gamma_{i,j}:=\partial\Omega_i\cap\Omega_j$ and $\Gamma_{j,i}=\partial\Omega_j\cap\Omega_i$ such that the problem defined in the overlap $\Omega_i\cap\Omega_j$ is uniquely solvable with the two type of conditions. And the matching conditions constitute a problem defined for the traces on the interfaces.

The Steklov-Poincaré operator is a general name referring to the maps between different types of traces. So, you are right that all the domain decomposition methods can be formed as solving an interface problem, and you can see the Steklov-Poincaré operators in the overlapping case has a little difference to the non-overlapping case: the former is defined between traces on different interfaces. For example, in the overlapping case, Dirichlet trace $u_i$ on $\Gamma_{i,j}$ can be used to solve out $u_i$ in $\Omega_i$ and then taking a Neumann trace $D u_i$ on $\Gamma_{j,i}\subset\Omega_j$ which should satisfy $D u_i = D u_j$; similarly, we have also an equation for $D u_j$ on $\Gamma_{j,i}$, which involves the following map: solving out $u_j$ in $\Omega_j$ using $D u_j$ and taking trace $u_j$ on $\Gamma_{i,j}$. The system looks like 2-by-2 blocks $$ \begin{pmatrix} I & -S_{i,j}\\ -S_{j,i} & I \end{pmatrix} \begin{pmatrix} u_i\\ D u_j \end{pmatrix}=.. $$ This is the same in appearance to the non-overlapping multiple-trace methods. While in the non-overlapping single-trace case, we have a single trace $d = u_i = u_j$ on $\Gamma_{i,j}=\Gamma_{j,i}$ and the interface system looks like $$ (S_{i,j}+S_{j,i})d = .. $$ where $S_{i,j}$ maps $d$ to another type (usually the dual type) trace through $\Omega_i$.

Answer 2

I'm not sure you need a Steklov-Poincare operator for overlapping domain decomposition methods. The Dirichlet-to-Neumann map (i.e. DtN or Steklov-Poincare operator) is useful in non-overlapping domain-decomposition because Neumann (i.e. flux) data is used in the transmission conditions between subdomains. A discrete DtN map just does this by solving the Dirichlet problem and computing the Neumann data from the solution.

Contrast this to, for example, additive Schwarz DD, where the solutions are simply added together at each iteration (communicating information through solution values and the overlap region). Similarly, for multiplicative Schwarz DD, the boundary condition on one domain is initialized with the value of the solution on the other domain. Neither one necessarily uses Neumann data at all. (I suppose the equivalent to the DtN map would just be the evaluation of the solution at the boundaries of the overlap region?)

Edit: there is some overlap, in that DtN maps have been used to define coarse spaces for DD methods.