Compatible Discrete Operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that approximate constitutive relations. We devise and analyze two families of such schemes for the Stokes equations in curl formulation, with the pressure degrees of freedom located at either mesh vertices or cells. The schemes ensure local mass and momentum conservation. We prove discrete stability by establishing novel discrete Poincaré inequalities. Using commutators related to the consistency error, we derive error estimates with first-order convergence rates for smooth solutions. We analyze two strategies for discretizing the external load, so as to deliver tight error estimates when the external load has a large irrotational or divergence-free part. Finally, numerical results are presented on three-dimensional polyhedral meshes.
In this paper we prove a complete panel of consistency results for the discrete de Rham (DDR) complex introduced in the companion paper [D. A. Di Pietro and J. Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes. Part I: Exactness and Poincaré inequalities, 2021, submitted], including primal and adjoint consistency for the discrete vector calculus operators, and consistency of the corresponding potentials. The theoretical results are showcased by performing a full convergence analysis for a DDR approximation of a magnetostatics model. Numerical results on three-dimensiona…
In this work we analyze a virtual element method on polyhedral meshes for solving the sixth-order elliptic problem with simply supported boundary conditions. We apply the Ciarlet-Raviart arguments to introduce an auxiliary unknown $σ:=-Δ^2 u$ and to search the main uknown $u$ in the $H^2\cap H_0^1$ Sobolev space. The virtual element discretization is well possed on a $C^1\times C^0$ virtual element spaces. We also provide the convergence and error estimates results. Finally, we report a series of numerical tests to verify the performance of numerical scheme.
Discrete Sobolev and Trudinger inequalities on polyhedral meshes
Univ. Lille, CNRS, Inria（里尔大学、法国国家科学研究中心、法国数字科学与技术研究所）；Institut de Mathématiques de Toulouse（图卢兹数学研究所；图卢兹大学；法国国家科学研究中心）；UMR5219 - Université de Toulouse（贡比涅技术大学）；CNRS - UPS；Université de Technologie de Compiègne
In this work, we develop a discretisation method for the mixed formulation of the magnetostatic problem supporting arbitrary orders and polyhedral meshes. The method is based on a global discrete de Rham (DDR) sequence, obtained by patching the local spaces constructed in [Di Pietro, Droniou, Rapetti, Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, arXiv:1911.03616] by enforcing the single-valuedness of the components attached to the boundary of each element. The first main contribution of this paper is a proof of exactness relations for this global D…
The purpose of the present paper is to develop $C^1$ Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We focus the presentation on the lowest order case, the generalization to higher orders being briefly provided in the Appendix. The degrees of freedom of the proposed scheme are only 4 per mesh vertex, representing function values and gradient values. Interpolation error estimates for the proposed space are provided, together with a set of numerical tests to valida…
We present the design of a mesh quality indicator that can predict the behavior of the Virtual Element Method (VEM) on a given mesh family or finite sequence of polyhedral meshes (dataset). The mesh quality indicator is designed to measure the violation of the mesh regularity assumptions that are normally considered in the convergence analysis. We investigate the behavior of this new mathematical tool on the lowest-order conforming approximation of the three-dimensional Poisson equation. This work also assesses the convergence rate of the VEM when applied to very general polyhedral meshes, inc…
Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes
De Vinci Higher Education（德文奇高等教育）；De Vinci Research Center（德文奇研究中心）；CNR IMATI（意大利国家研究委员会应用数学研究所）；Laboratoire de Mathématiques UMR 5127 CNRS, Université Savoie Mont Blanc（萨瓦省-蒙布朗大学CNRS第5127号数学实验室）；Department of Mathematics and Computer Science, University of Ferrara（费拉拉大学数学与计算机科学系）
This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e. in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the \textsc{Gaussian} divergence theorem then allows to analytically transform the volume integrals to curve integrals associated to the po…
In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of $L^2$-products, it can be used to design schemes for partial differen…
A simple but successful strategy for building a discrete diffusion operator in finite volume schemes of industrial use is to correct the standard two-point flux approximation with a term accounting for the local mesh non-orthogonality. Practical experience with a variety of different mesh typologies, including non-orthogonal tetrahedral, hexahedral and polyhedral meshes, has shown that this discrete diffusion operator is accurate and robust whenever the mesh is not too distorted and sufficiently regular. In this work, we show that this approach can be interpreted as equivalent to introducing a…
This paper analyzes the Scott-Vogelius divergence-free element pair on anisotropic meshes. Weexplore the behavior of the inf-sup stability constant with respect to the aspect ratio on meshes generated with astandard barycenter mesh refinement strategy, as well as a newly introduced incenter refinement strategy. Numericalexperiments are presented which support the theoretical results
Solutions to the wave equation in the exterior of a polyhedral domain or a screen in $\mathbb{R}^3$ exhibit singular behavior from the edges and corners. We present quasi-optimal $hp$-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an $hp$-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.
Design and homological analysis of twisted and BGG Stokes-de Rham complexes on polygonal meshes
IMAG, Univ. Montpellier, CNRS（蒙彼利埃大学 IMAG 研究中心）；School of Mathematics, Monash University（莫纳什大学数学学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the …
The Penalty Cell-Centered Finite Element Scheme For Stokes Problem On General Meshes
Faculty of Mathematics and Computer Science, University of Science, VNU HCMC（越南国立大学胡志明市科技大学数学与计算机科学系）；Faculty of Mathematics-Informatics, Ho Chi Minh City University of Pedagogy（胡志明市师范大学生物信息学系）
A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal-order error estimates are established for…
In this paper, a third order gas kinetic scheme is developed on the three dimensional hybrid unstructured meshes for the numerical simulation of compressible inviscid and viscous flows. A third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A simple strategy is adopted for the selection of big stencil and sub-stencils. Incorporate with the two-stage fourth-order temporal discretization and lower-upper symmetric Gauss-Seidel methods, both explicit and implicit high-order gas-kinetic schemes are developed. A variety…
A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids
In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth curved domains. This technique is based upon trial-functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast the test-functions are defined upon the standard degrees of freedom associated with the underlying method for …
We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier-Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier-Stokes…
The aim of this work is to introduce and analyze a finite element discontinuous Galerkin method on polygonal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poroelastic one. The coupling is realized by means of (physically consistent) transmission conditions, imposed on the interface between the domains, modeling different pore configurations. For the space discretization, we introduce and analyze a high-order discontin…
In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier-Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied without any postprocessing. The resulting method is pressure-robust so that the pressure approximation does not influence the velocity approximation. A new nonlinear convective term that earning non-negativity is proposed. The optimal convergence estimates for all the variables in $L^2$ norm are proved. Also, assuming that the rotational part of the forcing t…
Pressure-robustness has been widely studied since the conception of the notion and the introduction of the reconstruction approach for classical mixed methods in [5]. Using discretizations capable of yielding velocity solutions that are independent of the pressure approximation has been recognized as essential, and a large number of recent articles attest to this fact, e.g., [1,6]. Apart from the pressure-robustness aspect, incompressible flows exhibit anisotropic phenomena in the solutions which can be dealt with by using anisotropic mesh grading. The recent publications [3,4] deal with the c…
