A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing

We present a parallel data structure which is directly linked to geometric quantities of an underlying mesh and which is well adapted to the requirements of a general finite element realization. In addition, we define an abstract linear algebra model which supports multigrid methods (extending our previous work in Comp. Vis. Sci. 1 (1997) 27-40). Finally, we apply the parallel multigrid preconditioner to several configurations in linear elasticity and we compute the condition number numerically for different smoothers, resulting in a quantitative evaluation of parallel multigrid performance

Monolithic Multigrid for Magnetohydrodynamics

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems

Overlapping Schwarz Decomposition for Constrained Quadratic Programs

We present an overlapping Schwarz decomposition algorithm for constrained quadratic programs (QPs). Schwarz algorithms have been traditionally used to solve linear algebra systems arising from partial differential equations, but we have recently shown that they are also effective at solving structured optimization problems. In the proposed scheme, we consider QPs whose algebraic structure can be represented by graphs. The graph domain is partitioned into overlapping subdomains (yielding a set of coupled subproblems), solutions for the subproblems are computed in parallel, and convergence is enforced by updating primal-dual information in the overlapping regions. We show that convergence is guaranteed if the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap. Convergence results rely on a key property of graph-structured problems that is known as exponential decay of sensitivity. Here, we establish conditions under which this property holds for constrained QPs (as those found in network optimization and optimal control), thus extending existing work that addresses unconstrained QPs. The numerical behavior of the Schwarz scheme is demonstrated by using a DC optimal power flow problem defined over a network with 9,241 nodes

Multigrid methods for obstacle problems

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set

Ambiguity and Social Interaction

We examine the impact of ambiguity on economic behaviour. We present a relatively non-technical account of ambiguity and show how it may be applied in economics. Optimistic and pessimistic responses to ambiguity are formally modelled. We show that pessimism has the effect of increasing (decreasing) equilibrium prices under Cournot (Bertrand) competition. We also examine the effects of ambiguity on peace processes. It is shown that ambiguity can act to select equilibria in coordination games with multiple equilibria. Some comparative statics results are derived for the impact of ambiguity in games with strategic complements.

An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell's equations in waveguide configurations.The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell's equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations