72 research outputs found
Generalized Finite Element Systems for smooth differential forms and Stokes problem
We provide both a general framework for discretizing de Rham sequences of
differential forms of high regularity, and some examples of finite element
spaces that fit in the framework. The general framework is an extension of the
previously introduced notion of Finite Element Systems, and the examples
include conforming mixed finite elements for Stokes' equation. In dimension 2
we detail four low order finite element complexes and one infinite family of
highorder finite element complexes. In dimension 3 we define one low order
complex, which may be branched into Whitney forms at a chosen index. Stokes
pairs with continuous or discontinuous pressure are provided in arbitrary
dimension. The finite element spaces all consist of composite polynomials. The
framework guarantees some nice properties of the spaces, in particular the
existence of commuting interpolators. It also shows that some of the examples
are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8
figures and several comments adde
Nonlinear elasticity complex and a finite element diagram chase
In this paper, we present a nonlinear version of the linear elasticity
(Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric
embedding, metric, curvature and the Bianchi identity. We reformulate the
rigidity theorem and a fundamental theorem of Riemannian geometry as the
exactness of this complex. Then we generalize an algebraic approach for
constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes.
In particular, we discuss the reduction of degrees of freedom with injective
connecting maps in the BGG diagrams. We derive a strain complex in two space
dimensions with a diagram chase.Comment: Manuscript prepared for proceedings of the INdAM conference
"Approximation Theory and Numerical Analysis meet Algebra, Geometry,
Topology'', which was held in September 2022 at Cortona, Ital
Robust Preconditioners for Incompressible MHD Models
In this paper, we develop two classes of robust preconditioners for the
structure-preserving discretization of the incompressible magnetohydrodynamics
(MHD) system. By studying the well-posedness of the discrete system, we design
block preconditioners for them and carry out rigorous analysis on their
performance. We prove that such preconditioners are robust with respect to most
physical and discretization parameters. In our proof, we improve the existing
estimates of the block triangular preconditioners for saddle point problems by
removing the scaling parameters, which are usually difficult to choose in
practice. This new technique is not only applicable to the MHD system, but also
to other problems. Moreover, we prove that Krylov iterative methods with our
preconditioners preserve the divergence-free condition exactly, which
complements the structure-preserving discretization. Another feature is that we
can directly generalize this technique to other discretizations of the MHD
system. We also present preliminary numerical results to support the
theoretical results and demonstrate the robustness of the proposed
preconditioners
Complexes from complexes
This paper is concerned with the derivation and properties of differential
complexes arising from a variety of problems in differential equations, with
applications in continuum mechanics, relativity, and other fields. We present a
systematic procedure which, starting from well-understood differential
complexes such as the de Rham complex, derives new complexes and deduces the
properties of the new complexes from the old. We relate the cohomology of the
output complex to that of the input complexes and show that the new complex has
closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e
type inequalities, well-posed Hodge-Laplacian boundary value problems, regular
decomposition, and compactness properties on general Lipschitz domains.Comment: 31 pages. This preprint corresponds to the version accepted by
Foundations of Computational Mathematic
Nonstandard finite element de Rham complexes on cubical meshes
We propose two general operations on finite element differential complexes on
cubical meshes that can be used to construct and analyze sequences of
"nonstandard" convergent methods. The first operation, called DoF-transfer,
moves edge degrees of freedom to vertices in a way that reduces global degrees
of freedom while increasing continuity order at vertices. The second operation,
called serendipity, eliminates interior bubble functions and degrees of freedom
locally on each element without affecting edge degrees of freedom. These
operations can be used independently or in tandem to create nonstandard
complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show
that the resulting elements provide convergent, non-conforming methods for
problems requiring stronger regularity and satisfy a discrete Korn inequality.
We discuss potential benefits of applying these elements to Stokes, biharmonic
and elasticity problems.Comment: 31 page
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