37 research outputs found

    Virtual Element Methods Without Extrinsic Stabilization

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    Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension under the mesh assumption that all the faces of each polytope are simplices. The key is to construct local H(div)H({\rm div})-conforming macro finite element spaces such that the associated L2L^2 projection of the gradient of virtual element functions is computable, and the L2L^2 projector has a uniform lower bound on the gradient of virtual element function spaces in L2L^2 norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.Comment: 25 pages, 8 figure

    Novel H(symCurl)H(\mathrm{sym} \mathrm{Curl})-conforming finite elements for the relaxed micromorphic sequence

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    In this work we construct novel H(symCurl)H(\mathrm{sym} \mathrm{Curl})-conforming finite elements for the recently introduced relaxed micromorphic sequence, which can be considered as the completion of the divDiv\mathrm{div} \mathrm{Div}-sequence with respect to the H(symCurl)H(\mathrm{sym} \mathrm{Curl})-space. The elements respect H(Curl)H(\mathrm{Curl})-regularity and their lowest order versions converge optimally for [H(symCurl)∖H(Curl)][H(\mathrm{sym} \mathrm{Curl}) \setminus H(\mathrm{Curl})]-fields. This work introduces a detailed construction, proofs of linear independence and conformity of the basis, and numerical examples. Further, we demonstrate an application to the computation of metamaterials with the relaxed micromorphic model

    Complexes from Complexes: Finite Element Complexes in Three Dimensions

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    In the realm of solving partial differential equations (PDEs), Hilbert complexes have gained paramount importance, and recent progress revolves around devising new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as demonstrated by Arnold and Hu [Complexes from complexes. {\em Found. Comput. Math.}, 2021]. This paper significantly extends this methodology to three-dimensional finite element complexes, surmounting challenges posed by disparate degrees of smoothness and continuity mismatches. By incorporating techniques such as smooth finite element de Rham complexes, the t−nt-n decomposition, and trace complexes with corresponding two-dimensional finite element analogs, we systematically derive finite element Hessian, elasticity, and divdiv complexes. Notably, the construction entails the incorporation of reduction operators to handle continuity disparities in the BGG diagram at the continuous level, ultimately culminating in a comprehensive and robust framework for constructing finite element complexes with diverse applications in PDE solving.Comment: 46 pages, 6 figure

    New lower order mixed finite element methods for linear elasticity

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    New lower order H(div)H(\textrm{div})-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the (d+1)(d+1)-order normal-normal face bubble space. The reduced counterpart has only d(d+1)2d(d+1)^2 degrees of freedom. In two dimensions, basis functions are explicitly given in terms of barycentric coordinates. Lower order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient λ\lambda, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.Comment: 23 pages, 2 figure

    A lower order element for the linear elasticity problem in 3D

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    This paper constructs a lower order mixed finite element for the linear elasticity problem in 3D. The discrete stresses are piecewise cubic polynomials, and the discrete displacements are discontinuous piecewise quadratic polynomials. The continuity of the discrete stress space is characterized by moving all the edge degrees of freedom of the analogous Hu-Zhang stress element for P3P_3 [Hu, Zhang, Sci. Math. China, 2015, Hu, J. Comput. Math., 2015] to the faces. The macro-element technique is used to define an interpolation operator for proving the discrete stability

    A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

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    In this work we develop new finite element discretisations of the shear-deformable Reissner--Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M∈HZ⊂Hsym(Div)\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div}). The latter results in highly accurate approximations of the bending moments M\boldsymbol{M} and in the rotation field being in the discontinuous Lebesgue space ϕ∈[L]2\boldsymbol{\phi} \in [L]^2, such that the Kirchhoff-Love constraint can be satisfied for t→0t \to 0. In order to preserve optimal convergence rates across all variables for the case t→0t \to 0, we present an extension of the formulation using Raviart-Thomas elements for the shear stress q∈RT⊂H(div)\mathbf{q} \in \mathcal{RT} \subset H(\mathrm{div}). We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments M\boldsymbol{M}. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.Comment: Additional implementation material in: https://github.com/Askys/NGSolve_HuZhang_Elemen

    Mixed finite elements for Kirchhoff-Love plate bending

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    We present a mixed finite element method with parallelogram meshes for the Kirchhoff-Love plate bending model. Critical ingredient is the construction of appropriate basis functions that are conforming in terms of a sufficiently large tensor space and allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions, and polygonal convex or non-convex plates that can be discretized by parallelogram meshes, we prove quasi-optimal convergence of the mixed scheme. Numerical results for regular and singular examples with different boundary conditions illustrate our findings.Comment: corrected versio

    A robust lower order mixed finite element method for a strain gradient elasticity model

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    A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order C0C^0-continuous H2H^2-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.Comment: 25 page
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