37 research outputs found
Virtual Element Methods Without Extrinsic Stabilization
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 -conforming macro finite element spaces
such that the associated projection of the gradient of virtual element
functions is computable, and the projector has a uniform lower bound on
the gradient of virtual element function spaces in 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 -conforming finite elements for the relaxed micromorphic sequence
In this work we construct novel -conforming
finite elements for the recently introduced relaxed micromorphic sequence,
which can be considered as the completion of the -sequence with respect to the -space. The elements respect -regularity and
their lowest order versions converge optimally for -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
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
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
New lower order -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
-order normal-normal face bubble space. The reduced counterpart has only
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 , 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
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 [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
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
. The
latter results in highly accurate approximations of the bending moments
and in the rotation field being in the discontinuous Lebesgue
space , such that the Kirchhoff-Love constraint
can be satisfied for . In order to preserve optimal convergence rates
across all variables for the case , we present an extension of the
formulation using Raviart-Thomas elements for the shear stress .
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 . 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
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
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 -continuous -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