A Maximum Principle for L 2-Trace Norms with an Application to Optimized Schwarz Methods

Abstract

Summary. Harmonic functions attain their pointwise maximum on the boundary of the domain. In this article, we analyze the relationship between various norms of nearly harmonic functions and we show that the trace norm is maximized on the boundary of the domain. One application is that the Optimized Schwarz Method with two subdomains converges for all Robin parameters α > 0.

Efficient algorithms for solving the p-Laplacian in polynomial time

“…typically b = O( √ n) for d = 2 and b = O(n 2/3 ) for d = 3), one obtains an O(b 2 n) sparse matrix solve algorithm, resulting in O(n 2.5 log n) (d = 2) or O(n 2.84 log n) (d = 3) FLOPS for our overall algorithms. In addition, we mention many preconditioning opportunities [4,10,12,14,15,[22][23][24][25][26]35]. Although solution by preconditioning is possible, it is difficult to estimate the number of iterations a priori since the diffusion coefficient of the stiffness matrix is difficult to estimate a priori; in the best case ("optimal preconditioning") where the elliptic solve at each Newton iteration can be done in O(n) FLOPS, our algorithms are then O(n 1.5 log n) FLOPS.…”

Scharz Preconditioners for Krylov Methods: Theory and Practice

“…Scientists from five DOE labs presented work at the Twentieth International Conference on Domain Decomposition Methods in San Diego in 2011, and the meeting was in fact co-sponsored by Sandia and Livermore. We have made several contributions to the algebraic understanding of these preconditioners, and in some instances proved convergence results, or proposed new alternatives, which could not have been done without this algebraic approach [21], [22], [4], [23], [20], [34], [5], [32], [31], [6], [39], [35], [40], [1], [27], [29], [28], [30], [12], [24]. As one example, let us mention that the default Schwarz preconditioner in PETSc (from Argonne) is Restricted Additive Schwarz [7], and its convergence properties are understood thanks to our work [23].…”