HFSS15: Computational Time

A domain decomposition solution consists of three steps:

1. Mesh partitioning, a relatively fast process. HFSS will automatically partition the mesh into roughly equivalent sized mesh sub-domains to balance memory and solution times for the domains.

2. Matrix-assembly/solve where both A and M are solved in parallel

3. Domain iteration where we solve MAx=Mb via an iterative solver.

The solution time for solving the entire system (1) without DDM requires C₂N^b operations with a direct solver where C₂is a constant and b > 1 (and typically larger than a). For an M domain DDM solve, the factorization time of step 2 above is dominated by the most time consuming (i.e. largest) domains, given as:

Therefore, given b > 1, this stage exhibits a super-linear speed-up. This type of speed-up is commonly referred to as "scalable".

For step 3 the major computation is on the matrix-vector multiplication (MA)x which can be parallelized. In each iteration, the master node produces a new solution vector x and then sends its appropriate components to each domain (cpu/node/machine). Each domain then sends back to the master the vector y=MAx which is used by the master to produce a new solution vector x. This process goes on until x converges. In this step, the cost of matrix-vector multiplication per iteration is also scalable with respect to number of domains. Therefore if the iteration number remains relatively unchanged as number of domains increases, the domain decomposition solver can exhibit overall scalable performance.