Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.
We present a novel monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction (FSI) problem with a thick structure. A pressure-robust optimal energy-norm estimate is obtained for the semidiscrete scheme. When combined with a Crank-Nicolson time discretization, our fully discrete scheme is energy stable and produces an exactly divergence-free fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid (AMG) preconditioner.
We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming HDG schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique \cite{Xu96}. A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matri…
In this study, we present an $hp$-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The $hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equ…
We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0, for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the propo…
In [ESAIM: M2AN, 54(2020), 2229-2264], we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control {that reflects its global regularity}. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control proble…
We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes phase field model. The scheme is robust in the convection-dominated regime, produce a globally divergence-free velocity approximation, and can be efficiently implemented via static condensation. Two numerical benchmark problems, namely the rising bubble problem, and the Rayleigh-Taylor instability problem are used to show the good performance of the proposed scheme.
This paper analyzes a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the $\mathcal{P}_{k}/\mathcal{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, and piecewise $\mathcal{P}_k/\mathcal{P}_{k}$ for the trace approximations of the velocity and pressure on the inter-element boundaries. It is shown that the uniqueness condition for the discrete solution is guaranteed by tha…
A Reynolds-Semi-Robust, Globally Divergence-Free HDG Method for the Smagorinsky Model
Sichuan University（四川大学）；National Key Laboratory of Fundamental Algorithms and Models for Engineering Simulation, Sichuan University（工程模拟基础算法与模型全国重点实验室）
Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction
Sichuan University（四川大学）；National Key Laboratory of Fundamental Algorithms and Models for Engineering Simulation, Sichuan University（工程仿真基础理论及应用国家重点实验室，四川大学）
Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its orthogonal complement. Instead of characterization of these subspaces of the shape function space, characterization of the dual spaces are provided. Vector div-conforming finite elements are firstly constructed as an introductory example. Then new symmetric div-conforming finite e…
Two types of finite element spaces on triangles are constructed for div-div conforming symmetric tensors. Besides the normal-normal continuity, the stress tensor is continuous at vertices and another trace involving combination of derivatives of stress is identified. Polynomial complex, finite element complex, and Hilbert complex are presented and a commuting diagram between them is given. The constructed div-div conforming elements are exploited to discretize the mixed formulation of the biharmonic equation. Optimal order and superconvergence error analysis is provided. By rotation, finite el…
Several smooth finite element de Rham complexes are constructed in three-dimensional space, which yield three families of grad-div conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a cuboid. These elements naturally lead to conforming approximations to quad-div problems. Numerical experiments for each family validate the correctness and efficiency of the elements for solving the quad-div problem.
We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using H(div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the $L^2$-norm of order $O(h^{k+\frac12})$. We also prove error estimates for the pressure error in the $L^2$-norm.
Dipartimento di Matematica, Università di Milano（米兰大学数学系）；IMATI del CNR（意大利国家研究委员会应用数学研究所）；IUSS（高等研究大学）；Dipartimento di Matematica, Università di Pavia（帕维亚大学数学系）；Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca（米兰比可卡大学数学与应用科学系）
A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and …
This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H^2$ conforming finite element space, an $\boldsymbol{H}(\operatorname{curl}; \mathbb{S})$ conforming finite element space, an $\boldsymbol{H}(\operatorname{div}; \mathbb{T})$ conforming finite element space and an $\boldsymbol{L}^2$ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element…
Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I
Institute for Analysis and Scientific Computing, TU Wien（维也纳工业大学分析与科学计算研究所）；Institute for Numerical and Applied Mathematics, University of Göttingen（哥廷根大学数值与应用数学研究所）
Numerical analysis of a locking-free primal hybrid method for linear elasticity with $H(\mathrm{div})$-conforming stress recovery
Universidade Estadual de Campinas (Unicamp), Center for Energy and Petroleum Studies（坎皮纳斯州立大学（Unicamp），能源与石油研究中心）；Department of Applied Mathematics, Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学，数学、统计与计算机科学研究所，应用数学系）
In this paper, we propose a ${ P_{1}^{c}}\oplus {RT0}-P0$ discretization of the Stokes equations on general simplicial meshes in two/three dimensions (2D/3D), which yields an exactly divergence-free and pressure-independent velocity approximation with optimal order. Our method has the following features. Firstly, the global number of the degrees of freedom of our method is the same as the low order Bernardi and Raugel ($B$-$R$) finite element method (Bernardi and Raugel, 1985), while the number of {the non-zero entries} of the former is about half of the latter in the velocity-velocity region …
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
Finite element de Rham complexes and finite element Stokes complexes with various smoothness in three dimensions are systematically constructed. First smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. Based on the smooth scalar finite elements, both H(div)-conforming finite elements and H(curl)-conforming finite elements with various smoothness are devised, which induce the finite element de Rham complexes with various smoothness and the associated commutative diagrams. The div stability is established for the H(div…
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocit…
We consider a stabilization method for divergence-conforming B-spline discretizations of the incompressible Navier--Stokes problem wherein jumps in high-order normal derivatives of the velocity field are penalized across interior mesh facets. We prove that this method is pressure robust, consistent, and energy stable, and we show how to select the stabilization parameter appearing in the method so that excessive numerical dissipation is avoided in both the cross-wind direction and in the diffusion-dominated regime. We examine the efficacy of the method using a suite of numerical experiments, a…
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress $\bmσ=-\nabla^{2}u$ is sought in the Sobolev space $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ simultaneously with the displacement $u$ in $L^{2}(Ω)$. Stemming from the structure of $H(\bm{div},Ω;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\bm{div}\bmσ$…
The divergence-free velocity formulation of the consistent Navier-Stokes Cahn-Hilliard model with non-matching densities, divergence-conforming discretization, and benchmarks
