The swiss-carpet preconditioner: a simple parallel preconditioner of Dirichlet-Neumann type A. Quarteroni (Lausanne and Milan) M. Sala (Lausanne) A. Valli (Trento)

OUTLINE 1. Element-oriented and vertex-oriented decompositions 2. The swiss-carpet preconditioner 3. Analysis 4. Numerical results

1. Element-oriented and vertex-oriented decompositions Left: element-oriented Right: vertex-oriented

The model problem h is the grid size parameter

For the two-level Schwarz method, the construction of a coarse grid and of the corresponding restriction and extension operators can be computationally expensive on general unstructured grids. The vertex-oriented decomposition has been used in this framework for providing a coarse operator based on the so- called “aggregation”, avoiding the explicit construction of the coarse grid. A first result: the Schwarz algorithm on a vertex- oriented decomposition

For each subdomain one constructs a number of aggregates (sets of contiguous vertices). Each node of the fine grid is internal to only one aggregate (taking advantage of the vertex-oriented decomposition of the grid). The total number of the aggregates is the dimension of the coarse space. Only one coarse grid basis function is defined in each aggregate: the elements of the coarse space are given by linear combinations of elements of the fine space. The restriction operator is therefore a sort of piecewise constant interpolation. The numerical results have showed that this procedure is cheap and efficient (Sala and Formaggia, 2002; Sala, 2003).

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Algebraic structure for the element-oriented decomposition

Algebraic structure for the vertex-oriented decomposition

Some remarks:

2. The swiss-carpet preconditioner First of all, let us joke for a minute…: the reason of the name!

Start from the Swiss flag:

Modify the cross a little bit: Do you see the vertex-oriented decomposition?

Then take the Sierpinski carpet: the first four iterations are:

Here is the following one:

A Sierpinski-like procedure gives:

And here is the swiss-carpet!

The idea is to use a Dirichlet-Neumann preconditioner based on the swiss-carpet decomposition: this means one of the following choices P =

The first choice requires the solution of a Neumann problem in the red region (indeed, M independent Neumann problems in the M connected components of the red region) The second choice requires the solution of one Neumann problem in the white region

A third choice (a Neumann-Neumann preconditioner) is also possible: P = It requires the solution of M independent Neumann problems in the M red regions, and of one Neumann problem in the white region

In our experience, one finds better theoretical and computational results with the second choice, P =

The vertex-oriented decomposition is a suitable one for parallel computations: the nodes in the white transition region can be replicated on the processors which hold the disconnected red subdomains The Dirichlet step in the red region is clearly performed in parallel The vertex-oriented technique is used in many parallel linear algebra packages

3. Analysis The condition number of the preconditioned Schur complement matrix satisfies where the constant C does not depend on H and h.

The result is a straightforward consequence of the following estimate: that is obtained by carefully estimating the energy of the harmonic extension in the red regions by the energy of the harmonic extension in the white region.

Sketch of the proof. The first estimate is clear. The second is obtained by means of a frequently used approach: first, note that one can consider the local problem over a single red region and its white “frame”, and then add each local contribute; then, the harmonic extension in the red region is estimated by means of the trace over the boundary; by mapping back and forth on a square of unit side, the explicit dependence on H is determined (in this step, the estimate is uniform with respect to h, by the well- known uniform extension theorem);

finally, the trace over the boundary of the square is estimated by the energy of the harmonic extension (indeed, the interpolant) into the white frame (the only tool is the trace inequality); here, the frame is nonlinearly mapped into a “fixed” frame of width 1, and the explicit dependence on h and H is determined.

The result above has been proved for a second order symmetric elliptic operator having no zero-order terms, in the two- dimensional case (piecewise linear finite elements). However, numerical results seem to confirm it also for more general symmetric elliptic operators, and in the three-dimensional case.

Some remarks:

Generalized vertex-oriented decomposition One can use a larger “connecting” set: use for instance the following algorithm

4. Numerical results

The comparison between the EO Balancing Neumann- Neumann preconditioner and the VO swiss-carpet preconditioner

The comparison between the EO Balancing Neumann- Neumann preconditioner and the VO swiss-carpet preconditioner (continued)

We want to present now some additional numerical results for a non-structured subdomain decomposition The used decomposition is obtained by the software METIS METIS is a family of programs for partitioning unstructured graphs and meshes, and computing fill-reducing orderings of sparse matrices Other numerical results (work in progress)

Conclusions A simple preconditioner of Dirichlet-Neumann type has been presented Condition number (for the Laplace operator…) behaves like H/h (proved in 2D; numerical evidence in 3D) The main feature is the type of decomposition (the swiss- carpet, a suitable vertex-oriented decomposition) In principle, the same decomposition could be used for other problems (convection-diffusion, Stokes, systems,…) To be done: analysis and numerical results for more general problems