We consider the damped time-harmonic Galbrun's equation, which is used to model stellar oscillations. We introduce a discontinuous Galerkin finite element method (DGFEM) with $H(\operatorname{div})$-elements, which is nonconform with respect to the convection operator. We report a convergence analysis, which is based on the frameworks of discrete approximation schemes and T-compatibility. A novelty is that we show how to interprete a DGFEM as a discrete approximation scheme and this approach enables us to apply compact perturbation arguments in a DG-setting, and to circumvent any extra regularity assumptions on the solution. The advantage of the proposed $H(\operatorname{div})$-DGFEM compared to $H^1$-conforming methods is that we do not require a minimal polynomial order or any special assumptions on the mesh structure. The considered DGFEM is constructed without a stabilization term, which considerably improves the assumption on the smallness of the Mach number compared to other DG methods and $H^1$-conforming methods, and the obtained bound is fairly explicit. In addition, the method is robust with respect to the drastic changes of magnitude of the density and sound speed, which occur in stars. The convergence of the method is obtained without additional regularity assumptions on the solution, and for smooth solutions and parameters convergence rates are derived.
In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix-Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counte…
In this study, the nonconforming finite elements of order two and order three are constructed and exploited for the Stokes problem. The moments of order up to $k-1$ ($k=2,3$) on all the facets of the tetrahedron are used for DoFs (degrees of freedom) to construct the unisolvent $k$-order nonconforming finite element with the bubble function space of $P_{k+1}$ explicitly represented. The pair of the $k$-order element and the discontinuous piecewise $P_{k}$ is proved to be stable for solving the Stokes problem with the element-wise divergence-free condition preserved. The main difficulty in esta…
Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-co…
A Pressure-Robust Nonconforming Immersed Finite Element Method for Stokes Interface Problems
Jiangsu Key Laboratory of Quantum Computing Science and Devices, School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院量子计算科学与器件江苏省重点实验室）；Key Laboratory of NSLSCS, Ministry of Education, School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院教育部非平衡系统与复杂系统重点实验室）
Pressure-robust $hp$-a posteriori error estimates of $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin methods for the Stokes equations
Inria（法国国家数字技术研究院）；CERMICS, CNRS, ENPC, Institut Polytechnique de Paris（巴黎高等师范学院、国立信息与自动化研究所、巴黎理工学院）；School of Mathematical Sciences（东方理工学院数学科学学院）；Eastern Institute of Technology（香港城市大学数学系）；Department of Mathematics, City University of Hong Kong
Pressure-Robust Fortin-Soulie Elements of the Stokes Equation on Curved Domains
Peking University（北京大学）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）；Institut für Mathematik, Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconforming Helling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is $$ {\rm span}\left\{\left(\begin{array}{cc}1&0\\ 0 & 0\end{array}\right),\left(\begin{array}{cc}0&1\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}0&0\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}0&x\\ x & 0\end{array}\right),\left(\begin{array}{cc}0&y\\ y & 0\end{array}\right),\left(\be…
An interior penalty DG method with correct and minimal averages, jumps and penalties for the miscible displacement problem of nonnegative characteristic form, and SUPG-type error estimates under low regularity, dominating Darcy velocity
