Galerkin projection of discrete fields via supermesh construction

Interpolation of discrete FIelds arises frequently in computational physics. This thesis focuses on the novel implementation and analysis of Galerkin projection, an interpolation technique with three principal advantages over its competitors: it is optimally accurate in the L2 norm, it is conservative, and it is well-defined in the case of spaces of discontinuous functions. While these desirable properties have been known for some time, the implementation of Galerkin projection is challenging; this thesis reports the first successful general implementation. A thorough review of the history, development and current frontiers of adaptive remeshing is given. Adaptive remeshing is the primary motivation for the development of Galerkin projection, as its use necessitates the interpolation of discrete fields. The Galerkin projection is discussed and the geometric concept necessary for its implementation, the supermesh, is introduced. The efficient local construction of the supermesh of two meshes by the intersection of the elements of the input meshes is then described. Next, the element-element association problem of identifying which elements from the input meshes intersect is analysed. With efficient algorithms for its construction in hand, applications of supermeshing other than Galerkin projections are discussed, focusing on the computation of diagnostics of simulations which employ adaptive remeshing. Examples demonstrating the effectiveness and efficiency of the presented algorithms are given throughout. The thesis closes with some conclusions and possibilities for future work

Nervous Giant: China and the Financial Crisis

While the current financial crisis is widely acknowledged to be global, surprisingly little attention has been paid to its effect on one of the largest players in the global economy. China has weathered the crisis extremely well, though its growth has slowed slightly. I will analyze this by looking at China’s purchases of debt, the Chinese holdings of debt in the United States and its growing holdings in Europe, and the policy decisions directing this. This shows an intriguing change in the policy decisions that led to China becoming such a large holder of American debt. China amassed its large holdings of debt from the United States by merit of the strong trade relationship between the two countries, as well as the stability of the U.S. dollar. However, China’s interest in buying up Italian debt and forming stronger bonds with other Eurozone and European countries seems to speak to a different motive. Rather than allowing its reserves of foreign capital to grow over time, as it did with its U.S. debt, China is making a more aggressive move in this case. Thanks to its relative stability during the crisis, I believe this shows us that China is seeking to both ensure the continued security of its economic growth and increase its economic influence, thus using the instability of the global financial crisis to kill two birds with one stone

Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure