A Particle-Partition of Unity Method-Part III: A Multilevel Solver

In this sequel to part I [SIAM J. Sci. Comput., 22 (2000), pp. 853--890] and part II [SIAM J. Sci. Comput., 23 (2002), pp. 1655--1682] we focus on the efficient solution of the linear block-systems arising from a Galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method (PUM). We present a cheap multilevel solver for partition of unity (PU) discretizations of any order. The shape functions of a PUM are products of piecewise rational PU functions $\varphi_i$ with $\supp(\varphi_i)=\omega_i$ and higher order local approximation functions $\psi_i^n$ (usually a local polynomial of degree $\leq p_i$). Furthermore, they are noninterpolatory. In a multilevel approach we have to cope with not only noninterpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction. Hence, injection or interpolatory interlevel transfer operators are not available for our multilevel PUM. Therefore, the remaining natural choice for the prolongation operators are L2-projections. Here, we exploit the PUM construction of the function spaces and a hierarchical construction of the PU itself to localize the corresponding projection problem. This significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators. The second main ingredient of our multilevel solver is the use of a block-smoother to treat the local approximation functions $\psi_i^n$ for all $n$ simultaneously. The results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches $\card(\{\omega_i\})$. The convergence rate is slightly dependent on the local approximation orders pi.

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