We show that the gradient mapping of the squared norm of FischerBurmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newtons method.

In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the $j$th component, $c_j$, should be between 0 and 1. There are three main difficulties. Firstly, $c_j$ does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all $c_j's$ and enforce $\sum_jc_j=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to $c_j$. To construct the BP technique, we will not approximate $c_j$ directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

Surface segmentation is a fundamental problem in computer graphics. It has various applications such as metamorphosis, surface matching, surface compression, 3D shape retrieval, texture mapping, etc. All orientable surfaces are Riemann surfaces, and admit conformal structures. This paper introduces a novel surface segmentation algorithm based on its conformal structure. Each segment can be conformally mapped to a planar rectangle, and the transition maps are planar translations. The segmentation is intrinsic to the surface, independent of the embedding, and consistent for surfaces with similar geometries. By using segmentation based on conformal structure, the mapping between surfaces with arbitrary topologies can be constructed explicitly. The method is rigorous, efficient and automatic. The segmentation can be applied to surface morphing, construct conformal geometry image, convert mesh to Spline

In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, <i>SIAM Rev.</i>, 43 (2001), pp. 129--159]). The proposed error indicators are L_2-based and can be inexpensively computed, which makes our approach efficient

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