arXiv subjects

Yuanming Xiao

Publications and source records attributed to Yuanming Xiao.

4 recordsLinked to original sources

arXiv2025

An Unfitted Interface Penalty DG--FE Method for Elliptic Interface Problems

We propose an unfitted interface penalty Discontinuous Galerkin-Finite Element Method (UIPDG-FEM) for elliptic interface problems. This hybrid method combines the interior penalty discontinuous Galerkin (IPDG) terms near the interface-enforcing jump conditions via Nitsche method-with standard finite elements away from the interface. The UIPDG-FEM retains the flexibilities of IPDG, particularly simplifying mesh generation around complex interfaces, while avoiding its drawback of excessive number of global degrees of freedom. We derive optimal convergence rates independent of interface location and establish uniform flux error estimates robust to discontinuous coefficients. To deal with conditioning issues caused by small cut elements, we develop a robust two-dimensional merging algorithm that eliminates such elements entirely, ensuring the condition number of the discretized system remains independent of interface position. A key feature of the algorithm is a novel quantification criterion linking the threshold for small cuts to the product of the maximum interface curvature and the local mesh size. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2021

Core mass function of a single giant molecular cloud complex with ~10^4 cores

Similarity in shape between the initial mass function (IMF) and the core mass functions (CMFs) in star-forming regions prompts the idea that the IMF originates from the CMF through a self-similar core-to-star mass mapping process. To accurately determine the shape of the CMF, we create a sample of 8,431 cores with the dust continuum maps of the Cygnus X giant molecular cloud complex, and design a procedure for deriving the CMF considering the mass uncertainty, binning uncertainty, sample incompleteness, and the statistical errors. The resultant CMF coincides well with the IMF for core masses from a few $M_{\odot}$ to the highest masses of 1300 $M_{\odot}$ with a power-law of ${\rm d}N/{\rm d}M\propto M^{-2.30\pm0.04}$, but does not present an obvious flattened turnover in the low-mass range as the IMF does. More detailed inspection reveals that the slope of the CMF steepens with increasing mass. Given the numerous high-mass star-forming activities of Cygnus X, this is in stark contrast with the existing top-heavy CMFs found in high-mass star-forming clumps. We also find that the similarity between the IMF and the mass function of cloud structures is not unique at core scales, but can be seen for cloud structures of up to several pc scales. Finally, our SMA observations toward a subset of the cores do not present evidence for the self-similar mapping. The latter two results indicate that the shape of the IMF may not be directly inherited from the CMF.

astro-ph.GA↗

arXiv2016

High-Order Extended Finite Element Methods for Solving Interface Problems

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results.

math.NA↗

arXiv2010

An unfitted $hp$-interface penalty finite element method for elliptic interface problems

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument.

math.NA↗