Before proceeding further with the analysis and proofs of the theorems, we give a summary of the known bounds in Table 3.1 for other WR algorithms to compare the effectiveness of the newly found DNWR and NNWR algorithms.
Table 3.1 states that NNWR converges in about half as many iterations as DNWR. However, in comparison to DNWR, the NNWR has to solve twice the number of subproblems (once for Dirichlet subproblems, and once for Neumann subproblems) on each subdomain at each iteration.
The convergence results given in Theorems 2.2, 2.3, and 3.1 are based on technical estimates of kernels arising in the Laplace transform of the DNWR and NNWR algorithms.
In order to use Lemma 4.2 in our analysis, we have to show positivity of the inverse transforms of kernels appearing in the DNWR and NNWR iteration.
We now prove the main convergence results for the DNWR and NNWR algorithms stated in Sections 2 and 3.
Proof of Theorem 2.2 for DNWR. The goal is to obtain [L.sup.[infinity]] estimates for the DNWR error [w.sup.(k)] (t) in the case of [theta] = 1/2 and a > b, i.e., when the Dirichlet domain is larger.
Proof of Theorem 2.3 for DNWR. Recall that the recurrence in Laplace space reads
We perform experiments to measure the actual convergence rate of the discretized DNWR and NNWR algorithms for the problem