We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div-curl form using the nonconforming Crouzeix-Raviart space. Our main result is that the finite element method converges to a weak solution. The main challenge is to demonstrate the strong convergence of the density approximations, which is mandatory in view of the nonlinear pressure function. The analysis makes use of a higher integrability estimate on the density approximations, an equation for the "effective viscous flux", and renormalized versions of the discontinuous Galerkin method.
Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations
将Del Rey Fernandez et al.（2019）的双曲守恒律的熵守恒、曲线、双曲、p-精化算法从可压缩Euler方程推广到可压缩Navier-斯托克斯方程。采用了一种简单而灵活的耦合方法，相邻单元之间采用平面插值算子。通过最小化过程数值近似了曲率体积度量项，并满足离散几何守恒律条件。不同的曲线表面度量用于相邻的接口，以构建接口耦合项，从而本地化的离散几何守恒律约束到每个单独的元素。所得到的方案是熵保守/稳定的，元素保守的，和自由度保持。粘性界面耗散算子的开发，保持了熵的稳定性的基础计划。在粘性激波问题、雷诺数Re= 1，600时的泰勒-格林涡问题和Re = 2，000时的亚音速湍流流过球体的情况下，所得到的数值格式的精度和稳定性与原来的协调格式（达到~p+1收敛）相当。
Entropy Stable h/p-Nonconforming Discretization with the Summation-by-Parts Property for the Compressible Euler and Navier-Stokes Equations
本文将Del Rey Fernandez等人在曲线网格上求解可压缩Euler和Navier-斯托克斯方程的熵守恒/稳定算法推广到h/p加密/粗化。开发并行计算算法的主要困难是在并行计算接口之间建立适当的耦合过程。在这里，一个计算简单和有效的方法，使用解耦插值算子的基础上利用。所得到的方案是熵保守/稳定和元素保守的。等熵涡和粘性激波传播的数值模拟证实了该方法的熵守恒/稳定性和精度特性（达到~ p + 1收敛），与原始一致格式[4，35]的性能相当。对Re = 1，600时的泰勒-格林涡和Re = 2，000时通过球体的湍流的模拟显示了非结构网格的整体空间离散的鲁棒性和稳定性。最后，为了证明全离散显式熵稳定算法的熵守恒性质，我们采用松弛Runge-Kutta格式模拟等熵涡的传播，给出了该算法的熵函数随时间的演化.
We propose in this paper a discretization of the momentum convection operator for fluid flow simulations on quadrangular or hexahedral meshes. The space discretization is performed by the loworder nonconforming Rannacher-Turek finite element: the scalar unknowns are associated to the cells of the mesh, while the velocities unknowns are associated to the edges or faces. The momentum convection operator is of finite volume type, and its almost second order expression is derived by a MUSCL-like technique. The latter is of algebraic type, in the sense that the limitation procedure does not invoke …
We present a class of nonconforming virtual element methods for general fourth order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth order problems with varying coefficients. We also discuss fourth order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are…
We introduce and analyze a robust nonconforming finite element method for a three dimensional singularly perturbed quad-curl model problem. For the solution of the model problem, we derive proper a priori bounds, based on which, we prove that the proposed finite element method is robust with respect to the singular perturbation parameter $\varepsilon$ and the numerical solution is uniformly convergent with order $h^{1/2}$. In addition, we investigate the effect of treating the second boundary condition weakly by Nitsche's method. We show that such a treatment leads to sharper error estimates t…
We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconformin…
In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong conver…
We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the …
Despite the many advances in the use of weakly-compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence even for simple periodic domains. In this paper we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly-compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence despite having a periodic domain. Based on the study, we propose several new variations of t…
Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
We consider an alternative Navier-Stokes model for compressible viscous ideal gases, originally proposed in \cite{Svard18}. We derive a priori estimates that are sufficiently strong to support a weak entropy solution of the system. Guided by these estimates, we propose a finite volume scheme, derive the analogous estimates and demonstrate grid convergence towards a weak entropy solution. Furthermore, this existence proof is valid for "large" initial data and no a priori assumptions on the solution are needed.
This paper proposes a novel particle scheme that provides convergent approximations of a weak solution of the Navier-Stokes equations for the 1-D flow of a viscous compressible fluid. Moreover, it is shown that all differential inequalities that hold for the fluid model are preserved by the particle method: mass is conserved, mechanical energy is decaying, and a modified mechanical energy functional is also decaying. The proposed particle method can be used both as a numerical method and as a method of proving existence of solutions for compressible fluid models.
We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node-based uniform strain approach for finite elements. We refer to the proposed technique as the node-based uniform strain virtual element method (NVEM). No additional degrees of freedom are introduced in this approach, thus resulting in a displacement-bas…
In this paper, a piecewise quadratic nonconforming finite element method on rectangular grids for a fourth-order elliptic singular perturbation problem is presented. This proposed method is robustly convergent with respect to the perturbation parameter. Numerical results are presented to verify the theoretical findings. The new method uses piecewise quadratic polynomials, and is of the lowest degree possible. Optimal order approximation property of the finite element space is proved by means of a locally-averaged interpolation operator newly constructed. This interpolator, however, is not a pr…
