Multi-level partition of unity implicits
ACM SIGGRAPH 2003 Papers
A class of difference schemes with flexible local approximation
Journal of Computational Physics
Multi-level partition of unity implicits
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Journal of Computational and Applied Mathematics
Parallel generalized finite element method for magnetic multiparticle problems
VECPAR'04 Proceedings of the 6th international conference on High Performance Computing for Computational Science
A note on the conditioning of a class of generalized finite element methods
Applied Numerical Mathematics
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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.