In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C (H/h) and C (1+ log (H/h)) 2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant C depending on the infsup constant of the discrete spaces but independent of any mesh parameters. Here H/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates.

We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called StefanovUhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space. Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate the new method.

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In this work, an adaptive strategy for the phase space method for traveltime tomography (Chung et al 2007 Inverse Problems 23 30929) is developed. The method first uses those geodesics/rays that produce smaller mismatch with the measurements and continues on in the spirit of layer stripping without defining the layers explicitly. The adaptive approach improves stability, efficiency and accuracy. We then extend our method to reflection traveltime tomography by incorporating broken geodesics/rays for which a jump condition has to be imposed at the broken point for the geodesic flow. In particular, we show that our method can distinguish non-broken and broken geodesics in the measurement and utilize them accordingly in reflection traveltime tomography. We demonstrate that our method can recover the convex hull (with respect to the underlying metric) of unknown obstacles as well as the metric outside the

A BDDC (Balancing Domain Decomposition by Constraints) algorithm is developed and analyzed for a staggered discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems. On a quite irregular subdomain partition, an optimal condition number bound is proved for two-dimensional problems. In addition, a sub-optimal but scalable condition number bound is obtained for three-dimensional problems. These bounds are shown to be independent of coefficient jumps in the subdomain partition. Numerical results are also included to show the performance of the algorithm.