Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；Universität der Bundeswehr München（德国联邦国防军慕尼黑大学）；Universidad Nacional de General Sarmiento（圣马丁国立大学）；CONICET（阿根廷国家科学研究委员会）
我们在三角形的最大角条件以及该条件的两个推广（即所谓的三维最大角条件和正则顶点性质）下，证明了任意阶Raviart-Thomas插值的最优阶误差估计。我们的技术不同于以往关于该主题的论文中使用的技术，并且在几个方面得到更一般的结果。首先，允许中间正则性，即对于次数$k\ge 0$的Raviart-Thomas插值，当被逼近的向量场分量属于$W^{j+1,p}$时，我们证明了阶为$j+1$的误差估计，其中$0\le j \le k$，$1\le p \le\infty$，适用于三角形或四面体。这些结果即使在二维情况下也是新的。实际上，该估计仅在$j=k$的情况下已知。另一方面，在三维情况下，最大角条件下的结果仅对$k=0$已知。
英文摘要：
We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three dimensional maximum angle condition and the regular vertex property, for tetrahedra.
Our techniques are different from those used in previous papers on the subject and the results obtained are more general in several aspects. First, intermediate regularity is allowed, that is, for the Raviart-Thomas interpolation of degree $k\ge 0$, we prove error estimates of order $j+1$ when the vector field being approximated has components in $W^{j+1,p}$, for triangles or tetrahedra, where $0\le j \le k$ and $1\le p \le\infty$. These results are new even in the two dimensional case. Indeed, the estimate was known only in the case $j=k$. On the other hand, in the three dimensional case, results under the maximum angle condition were known only for $k=0$.
We investigate the piecewise linear nonconforming Crouzeix-Raviar and the lowest order Raviart-Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix-Raviart and the Raviart-Thomas finite-element approximate problems. We next present the equivalence between the Raviart-Thomas finite-element method and the enriched Crouzeix-Raviart finite-element method. We emphasise that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obt…
Identities that relate projections of Raviart-Thomas finite element vector fields to discrete gradients of Crouzeix-Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix-Ravaiart space onto elementwise constant functions are deduced.
We present precise Raviart-Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart-Thoma error estimates using the parameter. We also include corrections to an error in "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect.
The hybrid Raviart-Thomas mixed discontinuous Galerkin (HRTMDG) method is proposed for solving the Helmholtz equation. With a new energy norm, we establish the existence and uniqueness of the HRTMDG method, and give its convergence analysis. The corresponding error estimate shows that the HRTMDG method has an optimal $L^2$-norm convergence accuracy which is independent of wavenumber.
For the second lowest-order Raviart--Thomas mixed method, we prove that the canonical interpolant and finite element solution for the vector variable in elliptic problems are superclose in the $H(\text{div})$-norm on mildly structured meshes, where most pairs of adjacent triangles form approximate parallelograms. We then develop a family of postprocessing operators for Raviart--Thomas mixed elements on triangular grids by using the idea of local least squares fittings. Super-approximation property of the postprocessing operators for the lowest and second lowest order Raviart--Thomas elements i…
This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for t…
This paper considers the discretization of the Stokes equations with Scott--Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order $k$ of the Scott--Vogelius velocity space with appropriately chosen and explicitly given Raviart--Thomas bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 2021], where the case $k=1$ was studied. The proposed method is pressure-robust, wit…
Raviart-Thomas finite elements of Petrov-Galerkin type
Laboratoire de Mathématiques, Université Paris-Sud（巴黎南大学数学实验室）；Conservatoire National des Arts et Métiers（国立工艺学院）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142 Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学数学及其应用实验室）
Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix--Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Ravia…
In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first order mixed Raviart--Thomas element and the mixed Hellan--Herrmann--Johnson element, respectively. This, in particular, allows for proving a full one order superconvergence result for these two mixed finite elements. Finally, a full one order superconvergence result of both the Crouzeix-Raviar…
Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, w…
University Paris-Sud（巴黎第十一大学）；Conservatoire National des Arts et Métiers（法国国立工艺院）；Laboratoire de Mathématiques et de leurs Applications, CNRS, Université de Pau（波尔多大学数学及其应用实验室，法国国家科学研究中心）；Université de Pau（波尔多大学）
A Nonconforming Finite Element Method for the Biot's Consolidation Model in Poroelasticity
Tufts University（塔夫茨大学）；Universidad de Zaragoza（萨拉戈萨大学）；Penn State（宾夕法尼亚州立大学）；Institute for Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
The purpose of this work is to describe in detail the development of the Spectral Difference Raviart-Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the Flux-Reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows to build a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough a…
We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as $\mathcal O(h^{-1})$, while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as $\mathcal O(h^{-k-1})$, $k$ being the order of the Raviart-Thomas space. We rigorously prove that bo…
We revise the finite element formulation for Lagrange, Raviart- Thomas, and Taylor-Hood finite element spaces. We solve Laplace equation in first and second order formulation, and compare the solutions obtained with Lagrange and Raviart-Thomas finite element spaces by changing the order of the shape functions and the refinement level of the mesh. Finally, we solve Navier-Stokes equations in a two dimensional domain, where the solution is a steady state, and in a three dimensional domain, where the system presents a turbulent behaviour. All numerical experiments are computed using MFEM library,…
We present a class of discretisation spaces and H(div)-conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart-Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div)-conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general setting sh…
Recently, the $\vec{H}(\operatorname{div})$-conforming finite element families for second order elliptic problems have come more into focus, since due to hybridization and subsequent advances in computational efficiency their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems makes them interesting for a variety of applications, including incompressible fluids. In this area, boundary and interior layers are present, which demand the use of anisotropic elements. While for the Raviart-Thomas interpolation of any order on anisotrop…
We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS FE) method for scalar-valued convection-diffusion problems. The AVS-FE method is a Petrov-Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov Galerkin (DPG) method by Demkowicz and Gopalakrishnan, leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart Thomas trial spaces, th…
This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We then design a superconvergent recovery operator based on local weighted averaging. Combining the supercloseness and the recovery operator, we prove that the recovered flux superconverges to the exact flux. As a by-product, we obtain a superconvergent recovery estimate of the Crouzei…
We analyse the interpolation properties of 2D and 3D low order virtual element face and edge spaces, which generalize Nédélec and Raviart-Thomas polynomials to polygonal-polyhedral meshes. Moreover, we investigate the stability properties of the associated $L^2$ discrete bilinear forms, which typically appear in the virtual element discretization of problems in electromagnetism.
For the discretisation of $H_{div}$-functions on rectangular meshes there are at least three families of elements, namely Raviart-Thomas-, Brezzi-Douglas-Marini- and Arnold-Boffi-Falk-elements. In order to prove convergence of a numerical method using them, sharp interpolation error estimates are important. We provide them here in an anisotropic setting for the $H_{div}$-norm.
In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests.
We derive optimal and asymptotically exact a posteriori error estimates for the approximation of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for mixed eigenvalue approximations with Raviart-Thomas Finite elements. In addition, we use the post-processings introduced for the eigenvalue and eigenfunction based on mixed approximations with the Brezzi-Douglas-Marini Finite element. To combine these approaches, we define a novel additional local post-processing for the fluxes that appropriately modifies the …
For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $\mathbf A, \mathbf b,γ$ in $L^\infty$ and symmetric and uniformly positive definite coefficient matrix $\mathbf A$, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) finite element families of any order and in any space dimension and leads to the best-approximation estimate …
